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Ola Bratteli Derek W. Robinson
Operator Algebras and Quantum Statistical Mechanics 2 Equilibrium States. Models in Quantum Second Edition
Springer
Statistical Mechanics
Professor Ola Bratteli Universitetet i Oslo Matematisk Institutt Moltke Moes vei 3 1
0316 Oslo, Norway e-mail:
[email protected] Home page:
http://www.math.uio.no/~bratteli/
Professor Derek W. Robinson Australian National
University
School of Mathematical Sciences ACT 0200 Canberra, Australia e-mail:
[email protected] Home page:
http://wwwmaths.anu.edu.au/~derek/
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Second Edition 1997. Second
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2002
ISSN 0172-5998 ISBN 3-540-61443-5 2nd Edition ISBN 3-540-1038 1-3
Ist Edition
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SPIN 10885999
55/3141/ba
543210
To
Trygve Bratteli, Samuel Robinson, and Harald ROSS
Preface to the Second Edition
Fifteen years have and much has ments
passed since completion of the first edition of this book happened. Any attempt to do justice to the new develop-
would necessitate at least
edition of the current
one
new
volume rather than
a
second
Fortunately
other authors have taken up the challenge of describing these discoveries and our bibliography includes references to a variety of new books that have appeared or are about to one.
appear. We consequently decided to keep the format of this book äs a basic reference for the operator algebraic approach to quantum statistical mechanics and concentrated on correcting, improving, and the
updating
material of the first edition. This in itself has not been easy and changes occur throughout the text. The major changes are a corrected presentation of Bose-Einstein condensation in Theorem 5.2.30, insertion of a general result on the absence of symmetry breaking in Theorem 5. 3. 3 3 A, and an extended
in
The discussion of
in Sects. 6.2.6 and
description of the dynamics of the A^-Fmodel phase transitions in specific models, 6.2.7, has been expanded with the focus shifted from
model to
genuine quantum situations
such
äs
the
Example
6.2.14.
the classical
Heisenberg
Ising
and X-Y
models. In addition the Notes and Remarks to various subsections have been considerably augmented. Since
our
interest in the
subject of equilibrium states and models considerably in the last fifteen years
statistical mechanics has waned
of it
VIII
Preface to the Second Edition
impossible to prepare this second edition without the and encouragement of many of our friends and colleagues. We are Support indebted to Charles Batty, Michiel van den Berg, Tom ter Eist, particularly
would have been
Jürg Fröhlich, Taku Matsui, Andre Verbeure, helpful advice, and we apollatter. We the often for are especially grateful to Aernout ignoring ogize for Werner Reinhard and Enter van counselling us on recent developments and giving detailed suggestions for revisions.
Dai Evans, Mark Fannes,
and Marinus Winnink for information and
Oslo and Canberra 1996
Ola Bratteh Derek W. Robinson
Contents Volume 2
States in
Quantum Statistical Mechanics
5.1. Introduction
5.2. Continuous
3
Quantum Systems.
I
5.2.1. The CAR and CCR Relations
5.2.2. The CAR and CCR 5.2.3.
States and
Algebras Representations
6 6 15
23
5.2.4. The Ideal Fermi Gas
45
5.2.5. The Ideal Böse Gas
57
5.3. KMS-States
76
5.3.1. The KMS Condition
5.4.
l
76
5.3.2. The Set of KMS States
1 12
5.3.3. The Set of Ground States
131
Stability 5.4.1. 5.4.2.
and
Equilibrium
Stability Stability
144
of KMS States
144
and the KMS Condition
176
X
Contents Volume 2
5.4.3.
Gauge Groups and Systems
the Chemical Potential
197
5.4.4. Passive
211
Notes and Remarks
217
Models of
235
Quantum Statistical Mechanics
6.1. Introduction
237
6.2
Quantum Spin Systems
239
6.2.1. Kinematical and
239
Dynamical Descriptions Equilibrium The Maximum Entropy Principle Translationally Invariant States Uniqueness of KMS States Nonuniqueness of KMS States
6.2.2. The Gibbs Condition for
261
6.2.3.
266
6.2.4. 6.2.5. 6.2.6.
286 306 317
6.2.7. Ground States
6.3. Continuous
338
Quantum Systems. II
353
6.3.1. The Local Hamiltonians
355
6.3.2. The Wiener 6.3.3. The 6.3.4. The
Integral Thermodynamic Limit. Thermodynamic Limit.
366 I. The Reduced
Density
Matrices
II. States and Green's Functions
381 395
6.4. Conclusion
422
Notes and Remarks
424
References
463
Books and
Monographs
465
Articles
468
List of
487
Subject
Symbols Index
499
Contents Volume l
Introduction
l
Notes and Remarks
C*-Algebras 2.1.
and
16
von
Neumann
Algebras
C*-Algebras
19
2.1.1. Basic Definitions and Structure
19
2.2. Functional and 2.2.1.
Spectral Analysis
2.2.3.
Approximate
Representations 2.3.1.
25
Resolvents, Spectra, and Spectral Radius Identities and
32
Quotient Algebras
and States
Representations
39
42 42
2.3.2. States 2.3.3. Construction of
25
'
2.2.2. Positive Elements
2.3.
17
48
Representations
54
XII
Contents Volume l
2.3.4. Existence of
Representations
2.3.5. Commutative
2.4.
von
Neumann
2.4.1.
58
C*-Algebras
6l
Algebras
Topologies
on
65
^()
65
2.4.2. Definition and of
von
Elementary Properties Neumann Algebras
75
2.4.4.
79
Quasi-Equivalence
of
Representation
2.5. Tomita-Takesaki Modular of
von
2.5.1.
Neumann cr-Finite
Theory
and Standard Forms
Algebras Neumann
2.5.3.
Algebras Group Integration and Analytic Elements for One-Parameter Groups of Isometries on Banach Spaces
2.5.4.
Self-Dual Cones and Standard Forms
von
2.5.2. The Modular
2.6.
7l
2.4.3. Normal States and the Predual
83 84 86
97
102
Quasi-Local Algebras
118
2.6.1. Cluster
1 18
2.6.2.
129
2.6.3.
Properties Topological Properties Algebraic Properties
2.7. Miscellaneous Results and Structure 2.7.1.
Dynamical Systems and Crossed Products Operator Algebras Weights on Operator Algebras; Self-Dual Cones of General von Neumann Algebras; Duality and Classification of Factors; Classification of C -Algebras
2.7.2. Tensor Products of 2.7.3.
Notes and Remarks
Groups, Semigroups,
3.1. Banacb 3.1.1.
3.1.2.
3.1.3. 3.1.4.
3.1.5.
136 136 142
145
152
and Generators
Space Theory
Uniform
133
Continuity Strong, Weak, and Weak* Continuity Convergence Properties Perturbation Theory Approximation Theory
157
159 161
163 183
189 198
Contents Volume l
3.2.
XIII
205
Algebraic Theory 3.2.1. Positive Linear
205
3.2.2. General
228
3.2.3.
3.2.4. 3.2.5. 3.2.6.
Maps and Jordan Morphisms Properties of Derivations Spectral Theory and Bounded Derivations Derivations and Automorphism Groups Spatial Derivations and Invariant States Approximation Theory for Automorphism Groups
244
259 263 285
Notes and Remarks
298
Decomposition Theory
309
4.1. General
311
Theory
311
4.1.1. Introduction 4.1.2.
4.1.3.
4.2.
315
Barycentric Decompositions Orthogonal Measures
333
4.1.4. Borel Structure of States
344
Extremal, Central, and Subcentral Decompositions
353
4.2.1. Extremal
353
4.2.2.
Decompositions Decompositions
362
Central and Subcentral
4.3. Invariant States 4.3.1.
4.3.2. 4.3.3
4.3.4
4.4.
367 367
Ergodic Decompositions Ergodic States Locally Compact Abelian Groups Broken Symmetry
386 400
416
432
Spatial Decomposition
433
4.4.1. General 4.4.2.
Theory Spatial Decomposition
and
Decomposition
of States
442
Notes and Remarks
451
References
459
Books and
Monographs
461
Articles
464
List of
481
Subject
Symbols Index
487
States in
Quantum Statistical Mechanics
5.1. Introduction
In this
chapter, and the following one, we examine various applications of C*algebras and their states to statistical mechanics. Principally we analyze the structural properties of the equilibrium states of quantum Systems consisting of a large number of particles. In Chapter l we argued that this leads to the study of states of infinite-particle Systems äs an initial approximation. There are two approaches to this study which are to a large extent complementary. The first approach begins with the specific description of finite Systems and their equilibrium states provided by quantum statistical mechanics. One then rephrases this description in an algebraic language which identifies the equili brium states äs states over a quasi-local C*-algebra generated by subalgebras corresponding to the observables of spatial Subsystems. Finally, one attempts to calculate an approximation of these states by taking their limit äs the volume of the System tends to infinity, the so-called thermodynamic limit. The infinitevolume equilibrium states obtained in this manner provide the data for the calculation of bulk properties of the matter under consideration äs functions of the thermodynamic variables. By this we mean properties such äs the particle density, or specific heat, äs functions of the temperature and chemical potential, etc. In fact, the infinite-volume data provides a much more detailed, even microscopic, description of the equilibrium phenomena although one is only generally interested in the bulk properties and their fluctuations. Examination of the thermodynamic limit also provides a test of the scope of the usual statistical mechanical formalism. If this formalism is rieh enough to describe phase transitions, then at certain critical values of the thermodynamic parameters there should be a multiplicity of infinite-volume limit states arising from slight variations of the external interactions or boundary conditions. These states would correspond to various phases and mixtures of these phases. In such a Situation it should be possible to arrange the limits such that phase Separation takes place and then the equilibrium states would also provide information concerning interface phenomena such äs surface tension. The second approach to algebraic statistical mechanics avoids discussion of the thermodynamic limit and attempts to characterize and classify the equili brium states of the infinite System äs states over an appropriate C*-algebra. The elements of the C*-algebra represent kinematic observables, i.e., observables at a given time, and the states describe the instantaneous states of the System. For a complete physical description it is necessary to specify the dynamical law
4
States in
Quantum Statistical Mechanics
governing the change with time of the observables, or the states, and the equilibrium states are determined by their properties with respect to this dynamics. The general nature of the dynamical law can be inferred from the usual quantum-mechanical formalism and it appears that there are various possibiUties. Recall that for finite quantum Systems the dynamics is given parameter group of *-automorphisms of the algebra of observables,
A^Tt(Ä)^e^^^Ae-^^^
by
a one-
,
where H is the
selfadjoint Hamiltonian operator of the System. Thus it appears dynamics of the infinite System should be determined by a continuous one-parameter group of *-automorphisms T of the C*-algebra of observables. This type of dynamics is certainly the simplest possible and it occurs in various specific models, e.g., the noninteracting Fermi gas, some of which we examine in the sequel. Nevertheless, it is not the general Situation. The difRculty is that a group of this kind automatically defines a continuous development of every state of the System. But this is not to be expected for general infinite Systems in which compHcated phenomena involving the local accumulation of an infinite number of particles and energy can occur for natural that the
certain initial states. Thus it is necessary to examine weaker forms of evolution. For example, one could assume the dynamics to be specified äs a group of of the von Neumann algebras corresponding to a subclass of C*-algebra. Alternatively one could adopt an infinitesimal description and assume that the evolution is determined by a derivation which generates an automorphism group only in certain representations. Fach of these possible structures could in principle be verified in a particular model by a thermodynamic limiting process and each such structure provides a framework for characterizing equilibrium phenomena. To understand the type of characterization which is possible it is useful to refer to the finite-volume description of equilibrium. There are various possible descriptions of equilibrium states, which all stem from the early work of Boltzmann and Gibbs on classical statistical mechanics, and which differ only in their initial specification. The three most common possibilities are the microcanonical ensemble, the canonical ensemble, and the grand canonical ensemble. In the first, the energy and particle number are held fixed; in the second, states of various energy are allowed for fixed particle number; and in the third, both the energy and the particle number vary. Fach of these descriptions can be rephrased algebraically but the grand canonical be the Hubert space of description is in several ways more convenient. Let states for all possible energies and particle numbers of the finite System, and H and N, the selfadjoint Hamiltonian and number operators, respectively. The Gibbs grand canonical equihbrium state is defined äs a state over ^(), or ^^(), by
automorphisms states
over
the
Tr^(.-/^^^)
^^-^(^)^Tr,M^)
'
Introduction
where K erator.
for all
H
5
// G [R, and it is assumed that e~^^ is a trace-class opH is lower semi-bounded and the trace-class property is valid
iiN,
Typically ß > Q. The parameters ß
and /i
to the inverse
correspond
temperature of
the System, in suitable units, and the chemical potential, respectively, and therefore this description is well-suited to a given type of material at a fixed
temperature. Now if the generalized evolution A G
^()
^
then the trace-class property of f
are
T,(^)
-
is defined
e^^^Ae-^^^
e~^^ allows ^
T
one
G
^()
by ,
to deduce that the functions
cDß^^(Ait(B]]
analytic in the open strip 0 < Im ^ < jS and continuous Strip. Moreover, the cyclicity of the trace gives
on
the boundaries
of the
^ß.M'^t(B]]\t^iß This is the KM S condition which will
role
=
o}ß^^(BA)
brieüy described throughout this chapter. we
.
in
Chapter l and which significance of this over ^^(), i.e., the
One
important uniquely determines the Gibbs state only State over ^^() which satisfies the KMS condition with respect to T at the value ß is the Gibbs grand canonical equilibrium state. This can be proved by explicit calculation but it will in fact follow from the characterization of extremal KMS states occurring in Section 5.3. It also follows under quite general conditions that the KMS condition is stable under limits. Thus for a System whose kinematic observables form a C*-algebra ^ and whose dynamics is supposed to be given by a continuous group of *-automorphisms T of ^, it is natural to take the KMS condition äs an empirical definition of an equilibrium play
an
condition is that it
state.
Prior to the analysis of KMS states we introduce the specific quasi-local C*algebras which provide the quantum-mechanical description of Systems of point particles and examine various properties of their states and representations. In particular we discuss the equilibrium states of Systems of non-interacting particles. This analysis illustrates the thermodynamic limiting process, utilizes the KMS condition äs a calculational device, and also provides a testing ground for the general formalism which we subsequently develop. In the latter half of the chapter we discuss attempts to derive the KMS condition from first principles.
5.2. Continuous
Quantum Systems.
I
5.2.1. The CAR and CCR Relations There are two approaches to the algebraic structure associated with Systems of point particles in quantum mechanics. The first is quite concrete and physical. One begins with the Hubert space of vector states of the particles and subsequently introduces algebras of operators corresponding to certain particle observables. The second approach is more abstract and consists of postulating certain structural features of a C*-algebra of observables and then proving uniqueness of the algebra. One recovers the first point of view by passing to a particular representation. We discuss the first concrete approach in this subsection and then in Section 5.2.2
we
examine the abstract formulation.
The
quantum-mechanical states of n identical point particles in the configuration space U^' are given by vectors of the Hubert space L~(R"^''). If the number of particles is not fixed, the states are described by vectors of the direct sum
space
^=@L\K"-) 77
i.e.,
\l/ {jA^''^}>o, where given by
sequences
of
norm
i/^
is
=
ll'Af
=
l'/'^'P
+
E
,
>0
\l/^^^
G
C,
!/^^''^
G
L'([R'^')
for
fdxr--dx\il^^"\x,,...,x)\^
/?
>
l, and the
.
n>\'^ There If
is, however, a further restriction imposed by quantum statistics. G 5 is normalized, then
i/^
dp(xi,...,Xn)
=
\il/^''\xi,...,Xn)\^dxi
-
dx^
is the quantum-mechanical probability density for \l/ to describe n particles at the infinitesimal neighborhood of the points ;ci x^ The normalization of \l/ , corresponds to the normalization of the total probability to unity. But in .
.
.
,
.
microscopic physics identical particles are indistinguishable and this is refiected by the symmetry of the probability density under interchange of the particle coordinates. This interchange defines a unitary representation of the permutation group and the symmetry is assured if the ij/ transform under a suitable subrepresentation. There are two cases of paramount importance.
Continuous
Quantum Systems.
I
7
\l/^"^
of each ij/ are Symmetrie under The first arises when the components Particles whose transform in this manner are of coordinates. states interchange called bosons and
said to
are
satisfy
Böse
statistics. The second
(-Einstein)
ease
anti-symmetry of the i/^^"^ under interchange of each pair of particles are called /ermzö/?^ and are said to satisfy Fermi (-Dirac} statistics. Thus to discuss these two types of particle one must examine the Hubert subspaces 5.^, of 5, formed by the ij/ {^ }n>Q whose sign). These components are Symmetrie (the + sign) or anti-symmetric (the subspaces are usually called Fock spaces but we will also use the term for more general direct sum spaces. To describe particles which have internal structure, e.g., an intrinsic angular momentum, or spin, it is necessary to generalize the above construction of corresponds
to
coordinates. The associated
=
-
Fock space. Assume that the states of each
particle form a complex Hubert space l) and f) denote the /7-fold tensor product of i^ with itself. Fur ther introduce the Fock space g(l)) by
let
l)''
=
t)
0
I)
0
(g)
5(1))
©
=
t)"
,
n>0
if
where
i/^^"^
C. Thus
a
\\j
vector
G
is
5(f))
a
{iA''"''}/2>o
sequence
^^ vectors
subspace of g(l)) formed by the vectors with all components except the th equal to zero. In Order to introduce the subspaces relevant to the description of bosons and fermions we first define operators P on (5(^) by G
t)'^ and l)"
can
^+(/i
for all
be identified
äs
/.)
=
P-(fl^f2^"'^ fn]
=
/2
/l, ...,/
(TII, 712,
,
7r)
G
^
^
f). The
sum
the closed
(n
!)"^ V /.,
(n !)
is
of the indices and n is
/.,
0
Z-^^
^
over
one
^ Bnfn, all
^
even
/.
TI;
(1.
,
fn.
permutations
if TT is
0
0
and minus
2,
one
Extension
...,)
F-^
if TC is odd.
l and by linearity yields two densely defined operators with ||P|| the P extend by continuity to bounded operators of norm one. The P+ and P_ restricted to i^", are the projections onto the subspaces of 1^" corresponding to the one-dimensional unitary representations n \-^ l and TC 8;^ of the permutation group ofn elements, respectively. The Bose-Fock space g^(t)) and the Fermi-Fock space (5_({)) are then defined by =
H-
S(^)=/'5(l)) and the
corresponding -particle subspaces I)'^ by I)^ on g(^) by
==
{)"
P
.
We also define
number operator N
D(N)
=
(lA; l
^
=
{^^"^}.>o, E^'ll^^'^^ll' '^>o
and
7v,A
=
{.AW}>o
<
+^|J
a
8
States in
Quantum Statistical Mechanics
for each
\l/ G D(N). It is evident that A^ is selfadjoint since it is already given in spectral representation. Note that e^^^ leaves the subspaces 5.t(i^) invariant. We will also use TVto denote the selfadjoint restrictions of the number operator to these subspaces. The peculiar structure of Fock space allows the amplification of operators on I) to the whole spaces 5.^(1)) by a method commonly referred to äs second qiiantization. This is of particular interest for selfadjoint operators and uniits
taries. If H is
selfadjoint operator
f),
on
one can
define Hn
on
I)'^ by setting HQ
=
Q
and
ffn(P(f\
^'"^
fn]]
=P\y^fl^f2^-'-^Hfi^-"^fn
for all
// G D(H), and then extending by continuity. The direct sum of the // is essentially selfadjoint because (1) it is Symmetrie and hence closable, (2) it has a dense set of analytic vectors formed by finite sums of (anti-) symmetrized products of analytic vectors of H, The selfadjoint closure of this sum is called the second quantization of H and is denoted by dY(H]. Thus
dY(H)
@Hn
=
.
;i>0
The
simplest example
one
then has
of this second
quantization
jr(i)=7v If u is
and
unitary, [/
is defined
Un(P(fl
/2
0
0
extending by continuity.
by L/o
^
fn)]
=
The second
given by choosing
//
=
H ,
.
H and
P(Ufl
=
is
by setting 0
^/2
quantization
0
0
Ufn)
of U is denoted
by r(f/),
where
r(u)
@u
=
.
n>0
Note that r (U) is
is
a
strongly
unitary. The
notation dY and P is chosen because if Ut unitary group, then
r(t/,) Next
we
=
e'^^
continuous one-parameter
wish to describe two
bosons and fermions,
respectively.
-
e'"^''(^)
.
C*-algebras of observables associated with Both algebras are defined with the aid of
particle "annihilaüon" and "creation" operators which are introduced äs follows. For each / e 1) we define operators a(f), and *(/), on 5(1)) by initially setting a(/).A(0) 0,a*(/)^() /, / e ^, and =
a(/)(/i f2---fn)
*(/)(/! /2---/)
=
=
n^'^(f, /i)/2
/3
(+l)'^V
/l-
=
/
,
Continuous
Extension
i/^^"^
l)\
G
by linearity again yields easily calculates that
I
9
if
densely defined operators and
two
one
||a*(/),AW||
||a(/)^Wi| <'/2||/||||^(")||, Thus
Quantum Systems.
a(f]
and
(+1)'/2|/||||^W||
<
have well-defined extensions to the domain
*(/)
.
D(N^f^]
of
N^f^ and
\\a*(fm<\\f\ ||(A^+l)^/Vll for all
\l/ G D(N^^^), where a^(f} adjoint relation
denotes either
a(f)
or
a*(/). Moreover,
one
has the
(a*(/)(p,iA) for all
(p,iA
a(f]
and
D(N^^^).
G
<2^(/)
on
(/)
Finally,
we
=
define annihilation and creation operators
the Fock spaces =
/'i
a(f)P^
((p,ß(/)iA)
5^(1)) by a
,
; (/)
=
fl* (/)P
P
.
The relations
(/)?, A) follow from the
=
(,* (/W,
corresponding
iii(/)'/'ii
relations for
(/)-(/)^,
a(f}
11/11 ii(A^
<
and
a;(/)=Pfl*(/)
a(f) leaves the subspaces g.j_([)) invariant. f^-^a(f] are anti-linear but the maps /i-^a(/) are The ==
physical interpretation of .), then Q corresponds
(l, 0, 0,
.
i)'^Vii
*(/). Moreover,
because
Q
+
.
these to the
Note that the maps linear.
operators is
the
zero-particle state,
following. the
vacuum.
Let
The
vectors
A identify particle
with elements of the in the state
(/)
=
;(/)"
one-particle
space
1^ and
hence
a^ (/)
"creates"
a
The vectors
/.
iA(/i,...,/.)-^('^0~'/'<(/i)--<(/.)^ =
P(f\'--fn)
-particle states which arise from successive "creation" of particles in the /,/_!, /l Similarly the a (/) reduce the number of particles, i.e., they annihilate particles. Note that if fi=fj for some pair /, j with are
States
l
.
.
.
,
.
iA_(/b...,/)-^-(/i^---^/) by anti-symmetry.
Thus it is
=
o
impossible to create two fermions in the same principle which is reflected by the operator
state. This is the celebrated Pauli
equation
a*_(/K_(/)=0
.
10
States in
Quantum Statistical Mechanics
This last relation is the
simplest
of the commutation relations which link
case
the annihilation and creation operators. One computes straightforwardly that
[
(/), a + (9)]=0
+
=
[<(/),<(ö)]
[fl+(/),a;((7)]=(/,ör)1
,
,
and
K(/),a-(^)}
0
=
K(/),al(ö)}
=
{.(/), al(./)} where are
we
have
used the notation
again
=
(/,^)1
{A, B}
called the canonical commutation relations
,
,
AB + BA. The first relations
(CCRs)
and the second the
ca-
nonical anti-commutation relations
(CARs). Although there is a superficial similarity between these two sets of algebraic rules, the properties of the respective operators are radically different. In applications to physics these differences are thought to be at the root of the fundamentally disparate behaviors of Böse and Fermi Systems at low temperatures. In order to emphasize these differences we separate the subsequent discussion of the CARs and CCRs but before the general analysis we give an example of the creation and annihilation operators for point particles. EXAMPLE 5.2.1. functions of
tisymmetric (
n -
If
l)
=
then
L~(U'),
5^(1))
{iA^"^}>o
consists of sequences
of
variables x/ G [R^' which are totally Symmetrie ( + sign) or totally ansign). The action of the annihilation and creation operators is given by
(a(/)A)'"'(^i,
(al(f]il,f\x,,.
.
.
.
.
.
.x)
,x]
=
=
(n
-'/2
l)'/'
+
fdxj{x)'^^"^'\x,x^,.
;^( l)'-'/(^,),A('-')(.x,,
.
.
.
.
.,x)
,.x,,
.
.
,
,x)
.
,
i=\
where i/ denotes that the i th variable is to be omitted. Note that
/->(/), are
anti-linear and
a(f)
=
fl
(x),
and
a*_^ (x),
j dxW)^(^)
and then the action of these fields is
(a
W,A)W(^i,
(a;(;c)iA)("'(xi,
.
.
.
.
/^i(/)
linear, respectively,
tributions, i.e., fields
.
.
,x)
=
,x)
=
,
one
may introduce
operator-valued dis-
such that
<(f)
Jdxf(x)al(x)
=
,
given by
(n+
l)'/2 ,/.("+ ')(x,^,,
-'/2
^(
.
.
.
,x)
,
l)'-'5(.v -.x,OiA'"-"(-X|,
.
.
.
,x,,
/=!
In terms of these fields the number operator A^ is A^-
the maps
äs
formally given by
[dxa-'^(x}a^(x)
.
.
.
.
,x)
.
Continuous Quantum
analyze
5.2.1.1 The CAR Relations. We next
the
Proposition
5.2.2.
space, and
a(f)
(5_(I)).
erators on
Let
be
l)
a''(g]
and
a
the
on
the Fock space the operators.
5_(^).
complex Hubert space, 5- (W ^^^ Fermi Fock corresponding annihilation and creation op-
IK/)II
11/11
=
ll*(/)ll
=
for all / G ]^, and hence a(f] and a* (g) have bounded extensions. T/'Q (1,0,0, .) and {/} is an orthonormal basis ofl), then =
..
---^a.,)^
Aa,, -/.)-*(/,) is
(3)
on
It follows that
0)
(2)
of the creation and
properties
annihilation operators obeying the CAR relations We simplify Dotation by dropping the suffix minus
1 1
I
Systems.
of J5_(I))
orthonormal basis
an
finite
subsets
The
of
set
{/,
\vhen
,
.
,/}
-
-
the
^'uns over
ö/{/a}. {(/), ö*(ö^);
bounded operators
l)}
9 G
is irreducible
on
5-(f)). PROOF.
One has
(1)
(a\f)a(f)f
=
a^(f){a(f), a'(f)]a(f)
\\f\^a^(f)a(f)
=
and hence
ll(/)ll' As
a(/) ^
=
0 for
IIK(/)(/))'ll 0
/ ^
one
This follows
easily
Let r be
a
ll/ll'i|a(/)f
.
=
11/11
=
ii*(/)ii
.
from the observation that
'A(/c<,,---,A.) (3)
=
condudes that
ii(/)ii (2)
il/ll'lk'(/M/)||
=
(!)'^'^-(/a,
=
-
bounded operator in the commutant of
{a(f).a'(g): f,g f)},
then
m9ß---,gßj) (rn, a(A) a(/, )a*(i,^, ) a*(3/jjn) (ra,n)(,A(/,,-..,/.J,^(9#p..., /?))
W/,, ,/.,), =
=
To establish the last
equality
In the ßrst
.
one
.
.
.
considers the three
both
case expressions separately. particles than the a* (g) create. In the second zero by complex conjugation. In the third case (/) and the desired equality follows once more. Thus T consequence of Proposition 2.3.8.
m,
more
.
.
cases n
are zero
m, n
both
<
m,
a(f)
and
n
=
annihilate
expressions are again a*(gß^)Q. is a multiple of Q (Q, rQ)1l and irreducibility is a
case
=
>
because the
12
States in
Quantum Statistical Mechanics
5.2.1.2. The CCR Relations. The main and bosons is the absence of a Pauli
qualitative difference between fermions for the latter particles. There is no bound on the number of particles which can occupy a given state. This is quantitatively reflected by the unboundedness of the Böse annihilation and creation operators. If, for example, il/^^^ is the /7-fold tensor product of / G I) with itself, then the annihilation operator satisfies
principle
||a(/)^W||='/2|hAW||||/|| (we
omit the suffix
plus
the
on
operators). This unboundedness leads
number of technical difficulties which can
tors
be
partially
a(f)
and
avoided
absent for fermions. These
are
by consideration
a
of bounded functions of the opera
a*(ö^).
It is convenient to introduce
a
family
of operators
large problems
to
n(/)
=
$(//'),
+
a*(/))
{
G
l)} by
,
then
n(/)
=
-/2-'/2(a(/)_a*(/))
,
Thus
a(/)
=
and the
2-'/2(
/n(/)),
+
a*(/)
=
2-'/2(
_
/n(/))
,
a(f} ß*(/) can be recuperated from the (/). Thus for functional purposes it suffices to examine the latter operators. Their basic properties are most easily examined on the subspace F(t)) c S+(t)) formed by the finiteparticle vectors, i.e., the ij/ {^A }>o which have only a finite number of and
=
nonvanishing components. Proposition 5.2.3.
Let l) be a complex Hubert space, 5_^(I)) the Böse Fock and and a'' (g) annihilation and creation space, a(f) operators satisfying the canonical commutation relations. Define ^ by
0(/) for
all
/
(1)
G
t). Itfollows
For
\\f,
each -
f\\
(2)
If
(3)
For each
Q
{$(/,)
=
.
-
(1)
/Gl^,O(/) z^ essentially selfadjoint on F(bi) and if 0, then ||
analytie
.
The operator
vectors. G
-^
-
=
To establish essential
^(/)V^"^
2-i/2((/)+^*(/))
that
(
=
-
O(/)
is
But
^Im(/,6')iA
densely defined,
and Symmetrie, hence closable. it suffiees to prove that ^(f] has a dense set of if then i/^^"^ G D(
selfadjointness
f)';+^ 0^1"^
=
.
iA^''^GV|,
Continuous
l|ß(/)^<"'ll then
<
(
i)'^'ll/''"'llll/ll,
+
lk'(/)^'"'ll
Quantum Systems.
<
(
I
13
1)'^'||^("
+
that
imply
||(D(/)"V^"^II
^'"/-(^ + m)^'^(n +
<
m
-
l)^/^
(n
l)^/^||iA^"^
+
Therefore 1/2
^iw/r*'-HZ^(^
m^lia,(/rv(")|| ^ 4r^ !
y-iX^iiL.(^^L^]
<
||/ini^()||
< +00
m
m>0
for all ^ e C. It vector for
The
follows that each
immediately
i//
in the dense set
F(l))
is
an
analytic
(/). follows because
continuity
llW/a)
-
^(/))^||
<
2-^/2||a(/,
-
fm
+
2-^/2||a*(/,
-
/)iA||
<2^/2||A-/l|||(7V+l)^/Vll linear spans of {$(/i ) ^(fn]^\ // e I), n > 0} and {a* (/i ) I), W > 0} are identical. But the latter is dense by the same arguments used in Proposition 5.2.2. This is immediate from the canonical commutation relations. (3) The
(2)
a*(/)Q; //
Next
we
henceforth
G
unitary groups generated by the operators O(/), but Symbol to denote the selfadjoint closure of the previous
consider the we use
this
>(/). Proposition
For each
5.2.4.
/
G
t) /e/
O(/)
denote the
selfadjoint operator
(1)
For each
unitary operator exp{/
denote the
W(f)
pair f.g^l), W(f)D(^(g))
W(fmg}W(fr (2)
For each
pair /,
ö' E
W(f)W(g) (3) (4)
-
^(g)
-
=
D(^(g)}
Im(/, g)^
It follows that
and
.
l) =
The set {W(f)- / I)} If\\f,-f\\^0.then
e-'^^(f'a^l^W(f + g] is irreducible
on
,
5^(1))
.
\\(W(f.]-W(f}m^^
/o;-a//.AeS+(I)) (5)
For each
f
e
I)\{0} \\w(f)-n
2
.
(1) Each \l/^ e F(I)) is analytic for O(/) and one can define ^(g]W(fY \l/y^ by power-series expansion. This expansion yields the identity
PROOF. on
=
14
States in
Quantum Statistical Mechanics
^(^)^(/)^Aa and
Im(/, ^)l)iA,
-
consequently
II^(^)^(/)*OA. But
W(fr(^(g)
=
F(I))
is
'A/^)ll
-
mg)(^.
<
-
^^ß)\\
|im(/,
+
for
O(6f). Thus if i// G )(>(öf)), one may choose the il/,^ such that ^ 2.nd ^(ö')'Aa 'Aa ^(^)iA- It immediately follows that ^(g]W(fY\l/^ converges and therefore r(/)V e /)(O(^)) and ^(^)r(/)*iAcc <3[>(^)^^(/)*'A- Thus D(O(^)) a core
^
-^
-^
is invariant under each
and
W(f)
'^(g)w(fr on
H^(/)*((3)
=
im(/; 9)1)
-
_D(<1)(6()). (2)
For
\l/
F(I))
one can
the invariance of
exploit
etc., and the closedness of each
(/)
to derive the
D((^(g)]
under
W(f + g]'
,
identity
jW(tf)W(tg]W(t(f + g)Y^
then
Integration
=
W(tf)[i^(f}, W(tg)]W(t(f + g)Y,li
=
-/?Im(/, g)W(tf)W(tg)W(t(f the
gives
W(f]W(g]W(f + gY Iteration of this
following identity =
^-il Jo
dt
equivalent
n)(
to the stated
between bounded operators
=
e-^^^^^'^^^/'~
result.
bounded operator which commutes with each
a
D(^(f))
C
and
^(f)Til/ T^(f]ilJ the infinitesimal generator of t G ^^W(tf]. so r commutes with a(f) in the same manner
for each
calculation used for the CARs in
5.2.2.
=
D(^(f]]
\jj
a(f)
irreducibility
Proposition
=
->
-
w(itf]^(f]w(itfr for all
[R. Thus the spectrum of
t G
unitary
group
W(tf}
and its
^(f)
=
^(/)
-
0 for all
i//
G
(?+(f))
IhAf Thus if
'([7r
+ ,
one
.
has
II ^(/)^
-
7i-])i//
=
-
n~
i/^withO
=
2
/^OA, ^W^A)(i
< e <
-
71/2
4-||^(/)-l)'AllVlhAf <2|l-cos| andhence
||^(/)-1]||
=2.
i//
G
the
same
F(!)) by
converges
^ii/f 1
J
For each
by
must be the whole real line. Now consider
f dE(l)e'^^-
=
then
(/) is i^(if)) and
W(f^]
spectral representation
W(tf)
+
follows
that
lV(f),
because
2-^/-(O(/)
But and
(4) F(I)) is a core of (D(/) and ||($(/a) ^(f}W\ Proposition 5.2.3. Therefore, it follows from Theorem 3.1.28 strongly to W(f]. It follows from part (1) that (5)
the
.
identity immediately yields
If r is
(3)
g))^l^
tlm(f,g)W(tf)W(tg}W(t(f + g)Y
W(f)W(g)W(f^gY which is
+
cos/)
.
Continuous Quantum
The operators W(f) introduced in the above Weyl operators and the commutation relations
W(f}W(g) are
called the
=
e-^^^^f^^^'^W(f -^ g]
Weylform of the
we
15
are
usually
called
e-^^^^^^^^W(g]W(f]
canonical commutation relations.
5.2.2. The CAR and CCR In the last subsection
-
proposition
I
Systems.
Algebras
derived the canonical commutation and anti-com-
Weyl operators associated with the complete this discussion we next examine the abstract C*-algebras generated by elements satisfying the CARs or the Weyl form of the CCRs. It will result that these algebras are uniquely determined by the appropriate form
mutation relations and constructed the
former. To
of commutation relation.
Again
we
divide the discussion into two separate parts.
Algebra. The foregoing Fock space construction established the existence of bounded operators satisfying the CARs. The next result characterizes the abstract properties of the C*-algebra generated by these operators. 5.2.2.1. The CAR
1,2, I) be apre-Hilbert space with closure 5 and let ^/, i C*-algebras generated by the identity D and elements ai(f], / G I), satisfying Theorem 5.2.5. Let
be
=
two
(1) (2) (3) for
all
/->fl/(/) is antilinear, {/(/), /(^)}-0 (/, g]^ {,(/), a,(gr} ,
=
f,ge^.i= L
2.
It follows that there exists
unique ^-isomorphistn
a
a :
^i
F-^
^2 such that
a(a,(f])=a^(f] for ^
all
/
2X(]^)
G
^.
Thus there exists
^(5) generated by
=
commutation relations
over
a
unique,
elements
up to
*-isomorphism, C*-algebra
a(f], satisfying
the canonical anti-
l).
Furthermore
(1) (2)
Mf)\\ Ifl)
is
the
(3) (4) (5)
=
n
\\f\\ for
all
f&\^.
dimensional, where
n
<-\-OQ, then
2"
C*-algebra of2" complex is separable if, and only if, f) ^(^) is simple. If U is a bounded linear operator operator satisfying x
^(f))
F*(/ + t/*K
-
?7*t/ + F*F
=
0
=
D
-
^(1^)
is
isomorphic
with
matrices. is
on
separable. I) and
V
[/F* -f Ft/*
,
t/t/* + FK*
,
a
bounded antilinear
16
States in
Quantum Statistical Mechanics
then there exists
uniqiie *-automorphism
a
of ^(I))
y
such that
y(a(f)]=a(Uf)+a\Vf] and in this
case
y-\a(f))^a(U*f)+a^(rf) PROOF.
(1)
position (2)
5.2.2. It is
The
proof that ||fl/(/)||
is identical to the
If
E).
f\^a(f)
4f l ,
.
.
.
,
/-z
proof given
in Pro
a
ef,*
=
\\f\\
direct consequence of the CARs. Assume that I) is finite dimensional, and let {f\
basis for
for k
=
.
are
=
operators
satisfying
a(/,K(/,),
e<';'
F,_,a*(/,),
4?
=
=
,...,//;}
be
an
orthonormal
the CARs, define
F,_,fl(/,)
,
=*(/*)(/.)
where k
Vk
=
ll(^-2a^(fMff)} /=!
and
we
use
{ejy }.y^i
2
the Standard notation ^^^ ^^
*(/)
families of mutually
==
<^(fT-
commuting
4'^s:!
=
^^,=.4)4f)
=
Moreover, they generate the
same
algebra
äs
I^ follows from the CARs that 2x2 matrix
^.'^
,
4f)4'the
a(f)
-u)-rn(^s?-4^)Vn \i^\
units,
since
-
)
Hence this
(2), and
algebra is the algebra of 2" x 2" matrices, and this establishes property proposition when I) is finite dimensional. l) is infinite dimensional, construct an orthonormal basis {/} for l). For
the
When
each finite dimensional subset of äs
||fl(/)||
=
11/11
unique
{/} we may construct matrix units äs above, and by continuity that the C*-algebra generated by the a(/) is fact, generated by an increasing net of füll matrix algebras.
it follows
and is, in Moreover, ^l(l)) -^(5). The separability Statement (3) is immediate from the construction of ^il(I)) above, and the simplicity follows from Corollary 2.6.19 or can be proved äs follows: since {(/), ß*(/)} ll/f D, it follows that 7r(fl(/)) ^ 0 for all representations of TT of ^21(1)) and hence TT is faithful by the uniqueness of ^l(l)). Hence *:ll(l)) has no non trivial closed two-sided ideals. Statement (5) follows by applying the first Statement of the Proposition on a\(f} (/), a^d] a(Uf] + fl*(F/) =
,
=
=
.
Note that from Statement (2) of the theorem and the construction of ^(I)) it follows that this algebra is a UHF algebra (see Example 2.6.12). The transformations described in part (5) of the theorem are often called
Bogoliiibov transformations. We next investigate local space I).
structure of the CAR
algebra ^21(1))
over a
Hubert
Continuous
Proposition let Ibe
a
5.2.6.
net
Let
I
17
algebra over a Hubert space I), and subspaces ofl), orderedby inclusion such that:
be the CAR
^(t))
of closed nonempty
IfM^I, there exists an N e If M ^N and M ^ K, there
(1) (2)
Quantum Systems.
I such that M I.N.
exists
an
L e I such that M _L L and
N.KCL.
l)-UMe/^-
(3)
91(1)) be the sub-C'' -algebra generated by {(/); / G M} for each (^(f)), {^(M)}^^^) is a quasi-local algebra in the sense of Definition 2.6.3, mth o(a(f]] -a(f] for all fei). Let
a^(M)
C
Then
MG/.
=
First, if MI
PROOF.
C Mj, we evidently have 5l(Mi) C 2I(M2), and, second, äs any approximated by finite polynomials in a(f) and a*(^), it follows from the relation ||fl(/)|| \\f\\ and assumption (3) that 5I(^) UMe/^(^)- Third, the ^(M) have a common identity by definition. Finally, let er be the unique *-automorphism of ^ such that a(a(f]] (/) for all / G . This automorphism exists by applying Theorem 5.2,5 part (5) io U 0. Then (7~ l and F i and (j(5I(M)) ^(M) for all MG/. Each element A G ^(M) can be uniformly ap proximated by a sequence of polynomials P in the a(f) and 0^(0) with /, ö^ G M. But
A G
^(f))
be
can
=
=
=
=
=
=
=
if ^ is even, it follows that the
(P
(j(P,i})/2
is
(P
+
ö-(P))/2
also converge to ^1
=
(.4
+
G(A)]/2.
But
polynomial in the a(f) and a*(g). By a similar reasoning odd elements can be approximated by odd polynomials. Hence it suffices to prove the commutation relations for polynomials. But noting that A~ and B commute if ^ and B anti-commute, these follow directly from the CARs. +
an even
E) L^(U'') and, for each bounded open set A C [R^ define C*-subalgebra generated by {a(f) : / G L^(A)}. It follows that the CAR algebra ^(l)) is a quasi-local algebra with respect to this generating net. In particular EXAMPLE 5.2.7.
^IA
äs
Let
=
the
^(W-U^A
.
A
We conclude
gebra
which is
operators
by mentioning an equivalent way of describing analogous to the description of the CCRs in
{O(/);/
One defines
a
G
the CAR al terms of the
1^}.
family
of elements
5(/)
=
B(f)] /
G
f) by
2-'/2(a(/)+a*(/))
then
a(f)=2-^f^(B(f)+iB(if}} i.e., the a(f)
can
be recovered from the
B(f).
{B(f), B(g}}
-
,
But it follows from the CARs that
Re(/, g]
and conversely the CARs follow from these latter relations. Thus prompted to study the seemingly more general problem of an algebra erated by elements B(f],f G //, satisfying
{B(f],B(g]}=s(f,g]
one
is
51 gen
18
States in
for all
f.g^H, where
bilinear form
Quantum Statistical Mechanics /f is a real vector space and ^ is a real positive H. If in this context / is any operator such that
over
s(Jf, g) one can
ß
-s(f, Jg),
-l
=
,
introduce annihilation, and creation operators
aj(f) and
=
2-'/2(5(/)
=
iB(Jf)), a}(g)
+
symraetric
=
aj(f)
2-^/^(B(g)
-
and
aj(g) by
iB(Jg)]
has
one
{aj(f),a:,(g)}=s(f,g]-ris(f,Jcj] etc. Thus in
comparison
real part of
(f,g)
with the
previous
,
discussion
s(f, g] corresponds to the imaginary part. Although this latter description seems more general we remark that if // is a real Hubert space and if s is the nondegenerate inner product on H, then a / with the above properties exists if, and only if H has even (or infinite) dimension. In this case ff is a complex Hubert space and s is the real part of the inner product. One has the identification and
s(f,Jg) corresponds
to the
(ll+Ü2)(^-;.LC^
+
A2/^
for all A/ G [R and (^ G //, and
(f.g]=s(f,g)-ris(f,Jg] The
/
{^b
n\i
be
can
^2->
constructed
}
^27
^^z
Finally
==
first
by
of ^ ^^d then
defining Jni
^/,
choosing J by -^i
=
it follows that there exists
a
orthonormal
basis
'
calculation shows that this
mation of the
aj(f] U
and =
,
unique *-automorphism
y(B(f}) simple
an
remark that if T is any real invertible operator such that
we
s(Tf,Tg)^s(f,g]
A
.
aj(g),
^
B(Tf)
y of '5.1 such that
.
corresponds
to the
Bogoliubov transfor-
with
2~\T-JTJ),
V
=
2-\T^JTJ)
5.2.2.2. The CCR
.
Algebra. Next we characterize the abstract properties of the C*-algebra generated by the Weyl operators. Although this CCR algebra has many properties analogous to the CAR algebra the lack of norm continuity of the map /t-^ W(f) alters properties of density, separability, and quasi-locality. We will sHghtly generalize the Situation described in Section 5.2.1.2 and examine a family of Weyl operators W(f] defined for elements/of a real linear space H equipped with a nondegenerate symplectic bilinear form fj, i.e., er is a map from H
x
H into [R such that
^(/, g) for all
f,
g e
H, and if
=
-ö-(6^, /)
Continuous Quantum
ff(/, g] for all
/
e
//, then
(^. For
g
Hilbert space and
to be
a
example, given by
a(f, g) and then
symplectic properties
defined
a
pre-Hilbert
0
=
could take T/ to be
one
a
complex
pre-
Im(/, g]
on
because if
slight
H and there exists
=
an
^'
~a(f,Jg),
space with scalar
is
a
nondegenerate
operator
J
on
=
er
-l
H with the
,
multiplication
and inner
product
by
(^1
+
Ü2)/
(f.g} and
is very
generalization
bilinear form
a(Jf,g) then H is
19
I
recuperates the CCR relations described previously.
one
fact, the
In
=
Systems.
clearly (T(f, g)
=
=
-
AI/
^2Jf
-f
^(L JQ]
+
l/
,
G
G //
ff^,
/
/,
^ G //
i^(f. 9).
,
,
Im(/, g).
sequentially complete with respect to the topology defined above with the / a properties exists if, and only if, H does not have finite by (T, odd dimension. To construct / one first uses a procedure similar to the GramSchmidt orthogonalization procedure to find elements {^/, rjj} in H spanning a dense subspace and such that Note that if H is
(j(^t.^j) ^(^(^n^j}=^ (T(^i, r]j) Then
one
In the
defines /
by J^i
following,
when
Hilbert space H, it is inner product.
symplectic by nonzero
(1) (2)
öij
,
c/ and extension by continuity. r]^ and Jrj^ about the CCR algebra over a complex pretalking =
=
always understood
Let H be
Theorem 5.2.8.
=
bilinear form
a
a
all
=
is the
er
imaginary part
of the
equipped with a nondegenerate 2l/, l, 2, be two C*-algebras generated H, satisfying z
=
Wi(f),f G w,(fr, Wi(-f) Wi(f)Wi(g) e--(/'9)/2f^.(/ + g) ßr =
=
a
/
G //.
all
unique *-isomorphism
a(fF,(/)) ^
that
real linear space
and let
elements
It follows that there exists
for
,
Thus there exists
^(H) generated by Weyl
a
=
a;
^11-^^2 such that
ff2(/)
unique,
operators
f,geH.
up to
*-isomorphism, C*-algebra
W(f).
Furthermore
(1)
^(0) all
=
D, W(f), is unitaryfor all / / G //.
nonzero
G
//, and
\\W(f) -^=2for
20
States in
(2) (3) (4)
Quantum Statistical Mechanics
If H
is nonzero, then
^(H) If
T is
is a
5I(//)
is
real linear invertible operator
a(Tf, Tg] for
all
f,
nonseparable,
simple.
g e H, then there exists
on
H such t hat
a(/, cj]
=
'""
uniqiie -aiitomorphism
a
y
of'^(H)
such
that
y(W(f)] PROOF.
It follows from property
W(f]W((^]
=
W(Tf]
of the
(2)
W(f]
=
.
W(f] =
that
W(0)W(f)
and
W(-f]W(f) As the
W(f)
implies
that
=
W(0)
=
W(f]W(-f]
concludes that
.
I^F(O) identity l and property (1) unitary. Next let US regard the linear space // äs a discrete additive abelian group. The W(f] give a unitary representation of H up to a phase, or multipHer b(f, g] exp{-/ö-(/, g] /l}. It is subsequently of importance that f^^Xg(f) b(f, g] is a are nonzero one
W(f]
is the
is
=
=
character of //, e.g.,
b(fi+f2.ö]=b(f,,g]b(f2^g]
.
OUT aim is to prove that the C*-algebras %,/ =1,2, are Consider the two representations R^, and R of H defined
*-isomorphic. on
l^(H) by
(Rh(g}F}(f)=b(f,g)F(f + g) and
(R(g}F)(f)=F(f + g) One calculates that ^/, is
.
unitary representation up to the multiplier b and Ä is a sense. (Note that W (g) Rij(g) defines a re and hence a CCR algebra exists.) We may assume that ^li and % are faithfully represented on Hubert spaces , and 2 and define new multiplier representations Wf x R on /"(//; /) / 0 /^(//). An element ij/ G /"(//; /) is a function over H with values in g,-, and we set a
unitary representation in presentation of the CCRs
the usual
=
=
mxR}(g)^)(f) Then Wj
that Wi
x
X
R is
R is
=
W,(g)i^(f^g)
.
multiplier representation of H with multiplier b, and we next show quasi-equivalent to R^. Define a unitary operator Uf on /"(//, /) by n
(^.'A)(/)
-
w,-(f)iKf)
and then calculate that
(C/,-(F;
X
R)(gW(f)
=
=
that is,
Wi(f)((Wi
R](g]ilf](f) W,(f)Wi(g],lj(f + g) X
=
b(f, g]W,(f + g)il^(f
=
b(J\ ö)(C/,iA){/
+
fif)
+
=
g)
((!,-
Rb(g))ü,^)(f)
,
Continuous
Ui(WiXR)(g)U^ where
{(Wi
is
H/
X
the
R) (g)] g
G
identity
be
Let
y.
on
^/ identity,
the above
//}. Then, by
^i^Rb(g)
=
Quantum Systems.
I
2l
,
the
C*-algebra generated by a *-isomorphism T
there exists
from 93l onto ^2 such that
i((W, xR}(g)} Hence if
we can
find
*-isomorphisms
Ti(Wi(g))
=
:
T/
(W2xR}(g)
%
(W,
=
S/ such that
\-^
X
R)(g)
the first Statement of the theorem follows. But,
,
setting
W
=
Wj, this
amounts to
that
showing
^;,('xÄ)(/,.) for all /l/ e on
.
C, //
e
//, and
>
72
/^(//), unitary equivalent
E^'^(/'-)
1. The
to the
representation W x Ris, via Fourier transform representation W x R on l~(H;9)] defined by
((WxR](g]^|^](^]
W(g]l(g]^|^(y:), i^H
=
and hence
Y,)4WxK)(fi)
sup
=
y:^-x(//)^(//)
7,^H
NO W, the set
{7^}
of ^ E
^
of the form
x,,(/) is dense in H, since it is
lg(fW(f]
=
=
=
%,/r of
subgroup
a
W(g]W(f]W(gY.
MW'XÄ)(/-)
=
Ä
with annihilator
sup
y]^-;(,(/-)^(y;-) z^
^e//
i=\
)V(^)*
sup ge//
It follows that there exists
a
zero.
Moreover,
Therefore
=
/
*-isomorphism
a :
^Ij
E^'-^w-) /=i
H^
^2 such that
<^l(/))-^2(/) for all
/
Let
G
//, and
US next
a
is
unique
since the
prove the four
^(/) generate % of 5I(/f) stated
properties
äs a
C*-algebra.
in the latter half of the
theorem. At the
(1)
^(0)
t
=
.
beginning
The CCRs
of the
now
proof we
established that each
W(f)
is
unitary
and
give
W(g)W(f)W(gY=e-^-^^^^^W(f) and hence the spectrum of
W(f) is invariant under rotations. Hence the spectrum is 2 by the spectral radius formula. ||^(/) 11|| Assume that ^(//) is separable and let {^/J>o be a countable dense seof elements of ^(//). Thus for each t e ^ and each / G // there must be an rit
equal to (2) quence
the circle and
such that
-
=
22
States in
Quantum Statistical Mechanics
\\W(tf) But if
/^
0 and t{
then
7^/2,
-
A,\\
\\W{t,f)-AJ<\, would
\
.
because
^ ^^^
^^
<
\\W(t2f)-AJ
imply
\\W((t,-t2)f)-i\\<2 in contradiction to the first Statement
proved above.
cannot be dense and this contradiction proves that
(3) orem
Let
be
71
there exists
R is not coimtable A^
nonseparable.
representation of 5I(//), then by the first Statement of the the*-isomorphism a from '^l(H} onto n(^l(H)) such that a(PF(/)) =
for all / G //. But then Hence the kernel of TI is zero. But n This follows
^2(/)
äs
is
a a
n(W(f}) (4)
But
^l(H")
by applying
(x
=
was
the
because
n
^l(//)
is
generated by
the
W(f).
arbitrary and therefore ^(H) must be simple. uniqueness Statement to W\ (/) W(f) and =
W(Tf].
-
As in the CAR case, the transformations described in part often called Bogoliubov transformations,
of the theorem
(4)
are
There is
gous to
an
^l(I))
=
Proposition H and let is
a
important difference between
this theorem and the
result. Theorem 5.2.5, for the CARs. There is
sponding
'21(5)
^nd, and in fact,
5.2.9.
^(M)
Let
be the
one
has the
no
corre-
Statement analo-
following
Situation.
^(//) be the CCR algebra over the real vector space C'-suba!gebra generated by {W(f}', / G M} where M
siibspace of H.
It follows that
^(M) if,
and
only if,
PROOF.
If M
M
=
=
^(H)
H.
7^ //, consider
the
representation of ^l(H)
on
l^(H)
defined
by
(W(g}F)(f)=b(f,cj)F(f^g), i.e., the representation used in the proofof Theorem 5.2.8. Assume ^- e C,^/ eM, then W (g)
-
5^ X,r(^,) )F)(f)= b(f^ /
/-l
Hence, if
F is
/
g) F(/ + ^)
-
V
e M
~
gi
>
g]F(f
+
cj,]
ll^li
F(f+ g) is orthogonal to each of the vectors /(-^ /?(/, gi orthogonality follows because if / + ^/ e supp F C M, M, and hence f + g ^M. Therefore
This =
-
and let
/=!
because the vector /^
f
g,
H\M
supported by M,
r(^)-^;,.fF(<7,-)y xF(/ + ^/).
J] ;,/.(/,
g e
^.nf^^^||^(,)-.i|>l
-
g)
then
Continuous
lV(g)i ^(M)
andhence
We
consider
now
Quantum Systems.
I
23
.
quasi-local
structure on the CCR
algebra.
Proposition 5.2.10. Let H be a linear space mth a nondegenerate symplectic bilinear form a, If N and M are subspaces of H, define N ^- M if and only if Q for all f ^ N and g ^ M. Let I be a net ofnonempty subspaces of (j(f^ g ) H, ordered by inclusion, such that: =
IfMel, there exists an N ^ If M A^N and M \.K, there
(1) (2)
I such that M I.N. exists
an
L e I such that M L
and
N.KCL.
(3)
//
Let
^(H}
=
UMe/^-
algebra over H, and let t((M) M} for each MG/. (^(H], {^(M]}^^f) is a quasi-local algebra be the CCR
generated by {W(f]\ f 777^/7 tion
2.6.3, mth
a
in the
sense
of Defini
l.
W(f]W(g) /,
sub-C^-algebras
This is immediate from the relation
PROOF.
for
be the
G
=
e-'^^^^'^^W(g)W(f)
g G //.
Although
this Statement
5.2.6, there is
Proposition
a
on
quasi-local
structure is similar to
Proposition phenomena described in following example.
distinction which arises from the
5.2.9 and which is illustrated
EXAMPLE 5.2.1 1.
Let H be the
subspace
by of
the
L-([R')
formed
by
the functions with
compact Support. Moreover, let
a(J] g) If A is
a
=
bounded open set of U^' and ^IA is the then (^(//), {^Aj^^^^v) is
{^(/);/ e^"(A)}
Definition 2.6.3. However, cause
of
Im(/, g)
Proposition
5.2.3. States and
^(//)
is
not
.
C*-subalgebra of 5l(//) generated by quasi-local algebra in the sense of the CCR algebra over L-([R'') beto equal a
5.2.9.
Representations
analysis of the GAR and CCR algebras with a discussion of properties of states and representations of these algebras. Most of these properties are related to the existence of creation and annihilation operators and this causes a distinction between the CARs and CCRs. The CAR algebra contains creation and annihilation operators but these operators can only be affiliated with special representations of the CCR algebra. We begin with a discussion of these representations and associated states. We concentrate on the CCR algebra over a pre-Hilbert space l) and eschew the case of a real vector
We continue the
various
24
States in
Quantum Statistical Mechanics
space with case
is
symplectic form. barely more general.
There is
generators
a
As
we
have
explained
in Section 5.2.2.2 this latter
certain arbitrariness in the definition of the CCR
^(/)
in
Proposition 5.2.4, i.e.,
algebra from the algebra by C*-subalgebra of the
could define this
one
e^^^\ In fact the taking other functions of >(/) than /l CCR algebra generated by {W(tf)\t e IR} where / e f) is a fixed nonzero element is isomorphic with the set of almost periodic functions on -^
[R
=
cr(>(/)),
and there is
continuoLis functions
inherent
no
reason
not to
operate with the
set of
[R
vanishing at infinity, or any other subalgebra of Ciy(U) which separates points of IR. A consequence of this analysis is that one is not too interested in general representations of the CCR algebra, but only representations where the generators of ^^ ^(^/) exist. These are the so-called regulär representations. A representation (, TI) of the CCR algebra 51(1)), over the pre-Hilbert space 1), is said to be regulär if the unitary groups t G U\-^n(W(tf)) are strongly continuous for all / e f). If TT is regulär, then one can introduce, on , the selfadjoint infinitesimal generators ^T^(f),f G f) of the groups t^^n(W(tf]), / G I), and then use these to define annihilation and creation operators. Similarly a state co over ^(l)) is said to be regulär if the associated cyclic representation (^^, n.o-, ^co) is regulär. Note that on
\\(n,,(W(tf)]-^]7L,,(W(cj))^,,\\^ 2 e-^^^'^^^^^^^(D(W(tf]] e^^^^^^^^^^a}(W(-tf)) =
and ^ G [R
hence ^
it
-
-
follows
easily
G)(W(tf)]
G
to denote the infinitesimal
with
that
is regulär if, and only if, / G i). We use the notation ^,o(f] unitary group Ti^,:,(W(tf]) associated
co
C is continuous for all
generator of the
regulär state. The simplest example of a regulär state is the vacuum, or no-particle state, Q= (1,0,0,...) G a
Fock state cDp defined
e?_^(f)).
One
by the easily calculates
that
cof,(W(f)) Even if
co
is
a
=
regulär state,
(Q, W(f)ü)
there
are
=
^-ll/ll'/4 .
certain technical domain
problems
which
complicate the introduction of annihilation and creation operators. These are handled, however, by the following quite straightforward result. Lemma 5.2.12.
l) and CD
Let
UI(f))
regulär state generator of the unitary
be the CCR
algebra
over
the
pre-Hilbert
space
5l(^). For each / G l) denote the infinitesimal group t\-^7i(,j(W(tf}} by )f,j(/). It follows that the operators {)c^(/), 0f^(//),/ G M} have a common dense set of analytic vectors for every finite-dimensional subspace MCI). Moreover the annihilation and creation operators defined for eachf G f) by a
over
/)(a,,(/)) =Z)((D,X/))n/)(a),,(//)) -D(/)) and
Continuous Quantum
a^(f) are
=
2-'/^($4/) + /(!)(//)),
<(/)
densely defined, closed, flco(/)*
ll(/)('/)'?'ll' all (p e
for
PROOF. an
We may
assume
O'"/'^"
=
=
2-1/2 ($4/)
I
25
.^_^(.y-))
_
^'^^
2|K(/)'?'lP
ll/ll'll'Pll'
+
r>(a(/)). that M is
a
operator R
an
{fj-J
=
l, 2.
..
,m}
b&
g^ by
on
^""''"^'"''^(^(E(^^-
^^'"'^^''^""''^''
^'1
Let
complex subspace.
orthonormal basis of M and define
Rn
^w(/)'
=
=
Systems.
+
'''>)/>
It follows from rotational invariance that Rn is
Moreover, if for i//
e
f^
independent of the choice of basis. 0 by a si/?j/^ then ||i/^,j i/^|| Proposition 2.5.22. But using
defines
one
ij/^^ by i//,,
milar argument to that used to prove
W(tfj)W((sj
+
itj]fj]
=
-^
-
e-'"'I^W((sj +
itj)fj)
+
s
and
a change of variable in the multiple integral defining Rn one calculates that i// is analytic for each O^^ (/)). Hence i/^ is analytic for each operator Ofj(/), with / 6 M. It immediately follows that ciio(f] ^nd ß*j(/) are densely defined on D(0,,(/))nD(0,,(z/)). Moreover, ..(/)* D <(/) Thus a,,(f] has a densely de fined adjoint which means that it is closable. Similarly a^(/) is closable. Now if
q) G
D(a,,(f]],
then
||cD4/)cp||2 + ||04^y>ll'
<(/))^ll' + IIK(/) ^IM/)(p|p + IK(/)(p||^ (IIK(/)
=
+
-
<(/))cp|p)/2
.
Moreover, by differentiation of the Weyl relations
(
=
(
lk(/)d' + ll/fll
=
Combination of these two identities
yields
ll^..(/)^ll' + ll^a.(//)^||' One
now
assume use
may
selfadjoint,
and
in
-
and
||(D4,/)(^-^J||^0.
ll/ll'lklP
and =
ij/^^
But
.
||a,,(/)(iA \l/^
to
iA,J||
-
->
deduce
and
0.
that
(D^//)
are
~^
~
A similar argument is valid for
flfo(/)*
=
^lj(f)-
a*j(/).
This will be deduced
äs a
corollary
of
5.2.15.
Our next aim is to establish criteria for
the Fock
-^
+
particular closed, and hence i/^ e jD(Oa;(/))n)(
It remains to prove that
Corollary
2||a,(/)(^||^
0, i/^, e /)K(/)), ||iA. ^All identity just derived with cp
D(aco(f))- Another flfo(/) is closed.
therefore
-
that
the
l|1?X/)('A-i/'JII-0 =
'll/lPikll'
gives
IKo(/)'pll'
But
+
representation, i.e.,
the
a
state to be normal with
defining representation
of 9t (i^)
on
respect
to
the Fock
26
States in
5-|_(f)).
spaces
^ of
algebra
Quantum Statistical Mechanics In Section 2.6.2
=^()
we
defined
a
state
co
to be normal if it is determined
by
over
an
irreducible sub-
density
a
matrix p in the
form
CD(Ä}=i:r^(pA)
.
It follows from Theorem 2.4.25 that this is
equivalent to the representation copies of the representation of ^ on . The condition of normality is of interest in applications because the normal states of the Fock representation correspond to convex combinations of finite-particle vector states. Thus the normal states are interpretable äs the states with a finite number of particles. If the algebra is quasi-local, then the locally normal states, i.e., the states which are normal in restriction to the local generating algebras, correspond to states for which the local Subsystems have a finite number of particles. Thus locally normal states can be used to describe infinite Systems of particles for which the overall density is finite. Basically the criterion for normality is the existence of a selfadjoint number operator but äs this operator is almost always unbounded some care has to be
(cü5
^oj5
^cü) being
a
direct
sum
of
taken in its definition. On Fock space the number operator A^ posed in terms of operators Nf of the form
Nf
a^(f)a(f)
=
and it is consistent with the
creation operators that
Moreover,
one
easily
Nf
=
-(g,f)a(f)
=
(f,g)a*(f)
interpretation
be decom-
.
It follows from the CARs and CCRs that these operators
[Nf,a(g)] [Nf,a*(g)]
can
of (/)
satisfy
the relations
,
and a* (g)
äs
annihilation and
the number of particles in the state
measures
/
G
l).
calculates that
(^,7V.A)
=
^(./.,]V/, a
for
D(N} where the sum is over an orthonormal basis of f). We will extend these relationships to other representations. Notice that 7V/ is bounded for the CAR algebra but unbounded for the CCR algebra; thus we must first give a suitable definition of 7V/ for the CCR algebra and then find an appropriate definition of the sum of the TV/. We will use ij/
G
form
quadratic
lowing A (p,
0^
techniques
sesquüinear G
to handle
both these
problems.
We need the fol-
facts.
D(t)
X
form
D(t)\-^t(cp, i/^)
t
G
over
Hubert
a
C where
D(t)
is
linear in (p and Unear in i//. The form is said to D(t) is dense, Symmetrie if
^((p, lA) for all (/),
l//
G
,
-
is
space
of
TÖÄT^
and positive if
?(iA,^)>0 for all
\l/
e
D(t).
Positive forms
are
a
function
and t is anti subspace be densely defined if its domain
a
automatically Symmetrie.
Continuous
t(il/)
=
sesquilinear form quadratic form determines each
with
Associated
This
t(\l/, \l/].
;(,p,^) A
form
positive quadratic
=
t
->
extension and this is the and
case
by
is
27
quadratic form polarization formula a
is said to be closed whenever the conditions -
-^
-
the
I
^f^(^ + //4
0, and (3) ^(lA, lAII (1) iA e D(t), (2) ||iA and ^(lA/z 0. Alternatively, t is said to lA) -
there
t
t
Quantum Systems.
that
lA
e
closable if it has
a
closed
lA.)
be
-
0
imply
if, and only if, the conditions lA
that
^
Z)(0
-^(0^ 11^ II
~~^
^
0 imply 0. ^(lAJ important because if Ms a densely defined, positive, closed, quadratic form, then there exists a unique positive selfadjoint operator T such
t(\lf^
-
Forms
that
D(t]
iA,J
^
->
are
=
D(ri/2)
and
^((p,iA) for all (p,
lA
^
^(0-
I^
=
(r^/V,^^/V)
particular
r((p,iA)-((p,riA) for all (p e
and
/)(0
hence there is
lA
^
^(^)-
one-to-one
a
This Statement has
correspondence
positive selfadjoint operators. In appHcations the most sensitive point
an
obvious
between forms
t
and
converse
of this type and
in the construction of
positive
op
erators from forms is the verification of closedness of the forms. But if iS is
closed operator with dense domain
D(S],
t((p,i^}
=
and
one
defines
by D(t]
t
D(S}
=
a
and
(S(p,Sil^),
positive, densely defined, and closed. If T is the associated positive selfadjoint operator, then T^/^ is nothing but the positive part of the S* S. polar decomposition of S, and T If co is a regulär state over the CCR algebra ^(l)) on a pre-Hilbert space I), it is evident that t is
=
one can
introduce closed annihilation and creation operators fla;(/) and a*^(f} by Lemma 5.2.12. Let co, / and A^co, / be the positive form and the
on ^ positive selfadjoint operator
defined
by aa}(f}, i.e.,
c.,/W-l|ßc.(/)^AIl', A^co,/ We take
TV^^, /
äs
check from the
=
iAe^K(/))
öco(/)*öö;(/)
,
.
number operator for the one particle state / G I). It is easy to form of the CCRs and from Lemma 5.2.12 that one has
Weyl
./W It is natural to
=
(\\<^c.(fm'
attempt
+
\\^c.(ifm'
-
to define a total number
ii/iPii^ii')/2
operator for the
state
co
by
summing the co, / over a complete orthonormal basis, but again the difficulty is whether the resulting quadratic form is closed. To handle this we first define an Order relation between
positive
forms
by
ti > ti if
^(^i)
C
D(t2)
and
28
States in
Quantum Statistical Mechanics
tl(^)>t2(
ij/
D(ti).
G
erators. If
This
T{ and Tj
ordering defines an ordering of positive selfadjoint opthe selfadjoint operators determined by the positive,
are
closed, densely defined forms t\ and 6, TI
write
we
>
T2
whenever t\ >t2
This Order relation is
a
.
generalization of the order for positive elements of a
C*-
algebra discussed
in Section 2.2.2 and has many properties in common with this latter order. In particular the same argument that established Proposition 2.2.13 shows that if T\ and TI
are
positive, invertible, operators
two
^1
>
^2
>
and
0 ,
then
(T.+^^r^ for all /l
0. In
>
>
(Ti^i^r^
particular this means that )(77^'^") C D(T~^^~] by the defiordering. Moreover, the discussion after Proposition 2.2.13
nition of the form
implies
that
(r2 for all 1
0. These
>
Lemma 5.2.13.
points
+
will be used in
Let t^ be
quadratic forms
on a
^i)''/'>(ri+;j)'^/'
a
proving
the
following
monotonically increasing and define t by
net
resiilt.
of positive,
closed
Hilbert space
D(t)
=
t(\lj]
=
l lA; lA sup
G
p|/)(4), sup
4(iA)
4(iA)
< -{-00
l ,
.
a
Itfollows that t is a positive, closed, quadratic form. If, moreover, t^^ and t are densely defined, then the corresponding positive selfadjoint operators T and T satisfy lim
\\(e^^^^
-
e^^^)ilj\\
-0
a
for
a//
lA
PROOF.
G
, iiniformly for
t
infinite
intervals
of
[R.
As the net is monotonic
t(\lj]
=
\\mt^(\lj]
.
a
Thus
t
is the limit of
quadratic forms positive.
and it follows
easily that it
is itself
a
quadratic
form. Moreover, it is Next
assume
1^,^
G
D(t], ||iA,^
-
i/^||
^
0, and for
e
>
0 there is
an
A^^ such that
Continuous
t(^n for all n,m> N^. Therefore
ta(^n -^]
<^ for all
tM^
-
^n]
^n]
+
i//,)
ty,(\J/,^
<
s
I
29
< ^
for all
a
^n]
^^(^J
and
t^ is
äs
closed, \l/
e
^(^a)
and
N,. But
>
n
^m]
-
Quantum Systems.
+
t,((^
'AJ
-
-
^
^^.(^
+
-
l^J
and hence
^aW<2ra(lAJ+2
.
Moreover sup
4(i/^)
<
2^(i/^J
-^
sup
+ 2e < +00
a
and therefore
i/^
e
But then
D(r).
^(A
-
^A)
ra(iA,
lA)
-
<
a
for all
n
>
N^ and thus t(il/^
i/^)
0, i.e.,
-^
t
is closed.
Now consider the second Statement. The forms ^a, and hence the operators T^ + are monotonically increasing. Therefore, the two families (T + 11)" and (Ta + H)"
monotonically decreasing. Thus, by Lemma 2.4.19, (T^ + 1)" positive operator R and (7^ + 11)"^'^^ converges strongly to which, by continuity, is ^^/^. It also follows from monotonicity
^~
strongly positive operator that R > (r + 1])~^
are
to
1,
converges
a
a
This bound has two useful consequences. First, it shows that R is invertible, and R~^ its inverse is automatically selfadjoint. Second, it implies that
1)1/2) cZ)(/?-i/2).
/)((7 +
Next introduce
densely defined operators B^
^^/^(T^
=
||5a^||'-((ra + 1)^/V,^(7^a + l)^/V) by monotonicity. bounded with
D((ra
+
H)^/^)
Hence 5
||5*||
has
for each
=
lim||5,((ra
a
dense, By_ and
i/^
G
D(t^),
\\B^\\
<
l
and
then
ll'All'
i.e., the
ränge of
^^/^
B^
is
lies in
+
1)-^/2-^^/2)^||
=0
a
5
weakly to zero weakly to 1. But
converges
5* converge
11(5: and in
D(R-^I^],
If
One also has
a.
lim||/?'/2(1 -Bim and hence H
<
bounded extension with
a
1. It follows that
<
H)^/".
+
-
1)^f
particular \\B*^\j/\\
lim?,(iA)
2\\^f
(^,Bl^|<]
-
\\\l/\\. Therefore,
-^
+
<
||iA||'
lim||(r,
=
a
on the ränge of R^^^. As this ränge is the bound ||5*|| < l gives
-
(Bl^, ^)
D(R-^'^],
if
ij/
+
1)'/Vll'
^
0
then
a
=
hm||5:Ä-'/2,/,||^
=
||Ä-'/2,A|p
a
which
simultaneously
^W
ButonehasZ)(7?-i/2) conclude that
(r
+
D)
establishes that
i/^
e
/)(/)
::^
+
ll'All'-ll(^ + ^)'/Vf-P"^/Vll' Z)((r +
H)^/^)
converges =
-
H)^/^)
+
C
C
D(7?-V2)
-^
to
.
.
(H + T^)~^
the T
and
and hence r + H -/^-^.Thuswe
(r + H) -ö(^) and i/^^
strongly
Finally, note that if i/^ (H + T)" V ^ and Tai/^Qj (H (1 4- ra)~^)(p Ti//. Therefore, Ta converge strongly to the group generated by :==
D((T
=
30
States in
Now let
regulär
US
state
Quantum Statistical Mechanics
return to the discussion of the number over
CD
the CCR
algebra
dimensional
complex subspaces
inchision. If
{fi}
is
F C
a
on
I) form
operator associated with
pre-Hilbert a
space
a
The finite-
I).
directed set when ordered
by
orthonormal basis for F, define
an
n
D(n,,p)=
D(a,(f))
f^F
and
/VFW
=
E 'V/, W
Ell^-(/'>^ll'
=
As any two orthonormal bases for F are related by a unitary matrix, it follows immediately that this definition of 77^0^ p only depends on co and F ,
The /7co^/7 form a monotonically increasing net of positive, closed, quadratic forms and their limit will be a positive, closed, quadratic form by Lemma 5.2.13. If this latter form is
densely defined,
then it
uniquely determines
a
selfadjoint
number operator N^o on ^. The following result shows that the existence of such an operator characterizes normality of co, The main bürden in the
proof properties
is the demonstration of of the number operator
Theorem 5.2.14.
Let
be
algebraic properties
on
of N^o
analogous
to
Fock space.
the CAR
algebra, or a regulär state {ßco(/); / W corresponding closed annihilation operators (aco(f} bounded is in the CAR case}. De nco(a(f)) fine a positive, closed quadratic forni. on ^,j by over
the CCR
CD
algebra,
a
state over
and let
denote the
^
=
n,,(\jj)
=SUp
/2oj,F(lA)
,
F
DM
=
(lA;^
e
^
p
D(a^(f}),
72,,
where the supremum is over all finite-dimensional The following conditions are equivalent:
(1) (2) (3)
is normal with respect to the Fock
CD
D(nco} D(naj)
Moreover
is dense in
^.
contains
vector
if these
a
conditions
satisfied,
<
+00}
,
subspaces ofl).
representation.
which is cy die
are
W
J
fei)
for 7i,^(^(h))
in
^^.
then
lim||(e^'^^---/^"OiA||-0 uniformly for t in finite intervals of U where N^o is the selfadjoint operator corresponding to the form n^o and Nf^^ p is the selfadjoint operator determined by the form Uf^^p, PROOF.
direct
(1)
sum
rr^
of
(3): copies
If
co
is normal then
of the Fock
by Theorem 2.4.26 (,,^, TT^J) is a countable representation since the Fock representation is
Continuous
irreducible. Let N^j
N
=
operator and choose ij/
orthogonal,
and
ij/
be
N
{'A}>o D(Nia). Thus
G
=
direct
a
of
sum
such that
Sw
^
i/^
Quantum Systems.
3l
of the Fock mimber
copies
E
I
the
D(N),
i/^
are
mutually
(.A,7V,,iA)-E(^'^^) n>0
\^ sup Z^
=
^
">0
by
direct calculation
on
^ ||a(/,)iAJ|'^sup ^ ||.(//)^f .
^^
^
{/,}CF
{//}CF
irreducibility of the Fock representation ij/^ ensure that ij/ is cyclic for 7rf,,(5I(^)).
Fock space. But the
orthogonality of the components Thus condition (3) is valid. and the
(1)
for the
comment on the easier case of the CAR
algebra.
Weyl algebra. Subseproof relies upon three observations which establish invariance of /)(/!(,;) under the Weyl operators and the commutation properties of the number operator NOJ determined by /lojWe next prove the
quently,
we
implications (3)
=^
(2)
=^
The
OBSERVATION l /'
1
u;;v/"" \{\\^<^(t~'(W(tfi]-^]m~ -<j
Y.
/z,,W^sup
^yy^iji)
\\nf^(i
-
^ F {fi}<^F^
^'
^
+
||7r,Xr^(^(/V/)-1))iAf
if-lC
'}
-'"''"''
is exactly the set of \l/ for which the supremum is finite. e^^^ is a continuous unitary proof relies on the fact that if U (t) Hubert space, then ijj e D(H] if, and only if,
and
D(naj]
The
group
on a
sup||r'(t/W-iMII<+) t
3.1.23.
by Proposition
||//^f
=
Using
sup
the
spectral representation
\\t-\U(t)
-
1)^i|-
=
\\t-\U (t)
one
then finds that
l),Ai|-
-
.
/O
/
Now
lim
of U
the supremum in the Statement of the observation is finite. In par-
assume
ticular
sup{||r'(7t((r(^/) for each
/
e
l).
Thus
-1)),AII' + r'K(fF(/r/) -1))^||-
i/.
e
D(4>(/)) r^D(^^(if}]
-
D(a,,(f))
=
lliAII-}
=
< +(^
>(/))
and
sup{||r'K(^('/) -^)m' ||r'K(fF(rt/) l))^ir- m-}/2 \im[\\t-\n^(W(tf) l))^f \\r\n,,(W(itf) 1)),Af ||^^|i'}/2 +
=
=
m^m' + \\^o.(ifm||/||
=
versely, \l/
-
l where the last step
that the form defined
same
+
-
whenever
the
-
-
iiiAihA uses
-
=
-
iia>(/)./'f
Lemma 5.2.12. This identification shows
the supremum is at least an extension of Uf^. But if, conis in the domain of oj, then ^ e D(^(jj(f)) nD(^a}(if)) for all / E ^ and
by
argument identifies the
two forms.
States in
32
Quantum Statistical Mechanics
OBSERVATION 2
n,,(W(cj))D(n,:,)=DM, n,(K,(W(g))i^) This follows
by
first
=
(lA)
+
the
Weyl
using
(.A,1',o('ö)iA)
llöll'll'/'llV2
+
.
form of the commutation relations to
es-
tablish that
lim\\r^(n,(W(tf)-^))n(W(cj))>^\\-
Thiis
=
\im\\r\n,(W(tf)
=
\\<S>,(f)<^-lm(f,gm'-
by Observation
1))!^
-
+
r'(e-''""(^'')
-
^)n,,(W(tf)W\-
.
l
n^,(n,(W(g)}i^)
=
,7,,(iA) +
+
(^A, cD,X/^)'A)
ll^ll'lhAllV^
+
lim(i//,(D,X/(^F-^))jA),
wliere
Y.
9F=
{/:} Biit
lows
\\^co(i(gF
-
g}W\~
<
(2/7aX'A)
+
/'(/" ^)
C F
W\~)\\9F
-
g\\~
o and the desired result fol
-^
immediately.
The
D(/7,,j).
implication (3) => (2) is a consequence of Observation 2 il/ e D(n(,j} is cyclic, then )(wco) contains the dense
Thus if
linear combinations of
T^ü)(W(f]]\l/,f
G
and the set
linearity of by finite
formed
I).
OBSERVATION 3
a^(f]D(Nll'~]CD(N,,,] and for
ijj
e
D(Nl^~) N,a,,(f]ilj
Observation 2
implies
=
a,,(f](N,-^)ilj
.
that
(Moc/), W(tf]il^]
=
(W(tfYcp,N,ilj]^t(W(tfYcp,^(if]ilj]
+?^(r(//)>,iA)ll/llV2 for q) G D(N(o) and with / replaced by
ij/ G D(Nj~]. Differentiation of this expression and a similar /'/ and the subsequent addition of the two results gives
(M.c/),..(/)i//)
(/)c/), (yv,o
=
because C
D(N,] C D(n,) C ^^.D(^,,(f)}. ^rD(afo(f]]. Thus one has (7V,c/),a,,(/).A)
((p,ß.o(/)(A^a.
-
for all cp G D(N(o} where we have used tinuity of the right-hand side in (p. Now
we are
prepared
to prove
(2)
But then
a*^(fY
^
(1)
^
-
t)iA)
(7V,,
-
oX/)-
one
-
1])i//
G
D(Na/')
=
D(n,,]
1)^A) The result follows
of the theorem.
by
con-
Continuous
Quantum Systems.
I
33
Let EO) denote the
spectral family of NCO and for simplicity set P;. 'ü;(( cxo,/]). f) the operator P^a,a(f]Px is a bounded operator from PX^OJ ^o ^/^w because ||aü;(/)iA|| < llA^w^^^AII- ßut P^A^co and P^A^r/; are bounded operators on P;.fy and P^/tü and hence Observation 3 and functional analysis of bounded operators For each
/
e
gives
F(N^)P,a^(f]Px for any bounded function F. In
P,a,,(f)P;F(N^
=
-
1)
particular
P;^-ia^(f)P;i=P,a^(f)P,. for all ^> L Therefore,
P,-ia^(f)Pi But NOJ is one
positive
and hence if
=
a,,(f}P;,
.
/lo denotes the greatest lower bound of its spectrum,
must have
co(/)/^;.o+i/2-0 Thus if
is
i/^o
a
unit vector in the ränge of
c.(/)'Ao for all
/
.
P;^4.i/2 =
,
0
I).
G
Next let
P^Q be the projection onto the cyclic subspace generated by application of a'^(f) to IJ/Q. It follows from aüj(/)'Ao ^ ^^^^ ^^e subrepresentation
the operators
of Tifa is unitarily equivalent unitary operator is given by
PII/Q'^CÜ
to
the Fock representation.
[/<(/!) <(/)"< where Q
=
(1,0,0,...)
e
5^(^).)
Now
subrepresentation TT^ (H P^,)na, that PI//,^ is unitarily equivalent to by complete induction. --
In the CAR
case
=*(/!)
-
we
and
^
'(/)
repeat the -
the Fock
(The implementing
same
(1 -P,/,Ja;
construction for the
to find
i/^j
a
G
^
such
representation. The proof is achieved
Observation l is irrelevant. Observations 2 and 3
are
replaced
by
c.K(/)'A) +/^a.(/)^)
n,^((p,a,a(f)il/) which
are
=
C.W
-
na,(al^(f](p,\l/]
computed before taking
argument is identical
=
2||a.(/)^|P -
+
11/11'
,
((/),a^(/)iA)
the limit in the definition of o;-
The rest of the
to the CCR case.
The final Statement of the theorem follows
directly
from Lemma 5.2.13.
One classic consequence of Theorem 5.2.14 is the
following:
5.2.15 (Stone-von Neumann uniqueness theorem). Let ^(I)) be algebra over a finite-dimensional Hubert space f). It follows that each regulär state CD over ^(I)) is normal with respect to the Fock representation and hence any regulär representation 0/^(1^) is a multiple of the Fock representation.
Corollary the CCR
34
States in
PROOF.
have
Quantum Statistical Mechanics
From Theorem 5.2.14 it suffices to show that the operators {a,o(f]', / ^ W dense domain in ^^. But äs I) is finite dimensional this follovvs from
a common
Lemma 5.2.12. We
also
this result to
complete the proof of Lemma 5.2.12, i.e., to show regulär CD and / G I). The restriction of TT^^ to the CCR algebra over C/ is a multiple of the Fock representation of this algebra by Corollary 5.2.15. Hence it is enough to prove that a(/)* ö!*(/) in the Fock representation. Writing the Fock space ^^ äs a direct sum of 7i-particle spaces that
can
afü(/)*
=
use
for sach
^l^(f]
=
5+-©f)+ know that
we
iy|
fl*(/), since it is fl*(/) C a(fY. To means
D(a*(f))
C
a
joint
for all
n
for
core
show the converse,
,
and the Hnear Span of the iy| is a core for >(/) and (//). We know already that assume
that
i//
=
0^^^^ i/^^"^
e
D(a(ff].
This
it
follows
from
that the functional
(peß(a(/))^(^,a(/),p) is
bounded.
(/)!)';
C
But
I,;-'
if
cp
@,j>o^/^^"^
=
is
finite-particle
a
vector
that
(iA,a(/)rp)
=
^(^(''-",a(/)
=
n>\
;^ (a* (/),A<"-'), (?<"') n
Hence the boundedness in cp
implies
>
.
l
that
^||'(/),^(")||-<+CX3
.
/z>0
But
äs
a*(f)
is
closed, this implies
that
i//
=
0>o^^"''
(a*(/)iA)(")^a*(/)^A("-^)
^
^('^*(/))
and
.
Thus
*(/)=(/)* The one
proof
employing rem
have given of existing proofs.
we
of several
von
One
topology
äs
the
.
Neumann
can
uniqueness theorem is only example give another proof by uniqueness of the CCRs in Theo
for
the argument used to show the
5.2.8, but regarding f)
usual
the
(/)-*(/)*
,
locally compact
rather than the discrete one, where
of
n
abelian group U~" in its complex dimension
is the
{). Corollary 5.2.15 identifies the regulär states on the CCR algebra ^(1^) over a general pre-Hilbert space l) with the states which are locally normal with respect to the quasi-local structure defined by the finite-dimensional complex subspaces of l). Thus, äs far äs regulär states are concerned, one could define the CCR algebra äs a closure of a union of algebras of the form ^(5+(^^)), where g_^ (M) is the Symmetrie Fock space over the finite-dimensional subspace M and ^(g_^(M)) is the weak closure of ^(M) in the Fock representation. This latter definition of the CCR algebra gives a larger algebra without the deficiencies mentioned in the introduction to this subsection, i.e., it contains all bounded Borel functions of the fields
O(/).
Continuous
Quantum Systems.
I
35
Although the Fock representation is the unique irreducible regulär representation of ^(C'), up to equivalence, it often appears in a seemingly different form, e.g., it occurs äs the Schrödinger representation in quantum mechanics. The physical interpretation of this latter reformulation is rather different and it is of some interest to cast the representation in this form because it also immediately allows the construction of a nonregular irreducible representation.
{ffj
Let
(The Schrödinger representation).
EXAMPLE 5.2.16
=
1,2, ...,}
be
orthonormal basis of the finite-dimensional Hubert space I) and introduce 2n one-parameter groups Uj and Vj by
an
Uj(sj) where the W
W(sjfj)
=
These groups
Weyl operators.
are
FXO)
,
=
W(itjfj]
can
be
represented
((/X.,)W(^i,-.-,^.)-^''^-^^>(^h...,^.) (^j(tj}^)(x\^"-^Xn} Now if
/
=
,
on
L- (U")
by
,
ll/(X],...,Xj-tj,...,Xn)
.
l) has the decomposition
e
/^^(.,
/0-)/y
+
,
7=1 one
obtains
representation
a
of the
Weyl algebra
with
lV(f}=l[e'^''^''-Uj(sj)Vj(tj)
.
y=i
In
particular
('^(.fj)i'}(x\,.--,x) =Xj\l/(x\,...,x) ((D(;/;)iA)(^l
,
.
.
,xn)
.
=
-i^^(xi,. dXj
,
.
,x)
.
and
.(/,)
=
2-"=(,, |-), .,/,)^2-(.,-A). +
Now the Fock vacuum,
i.e., the
H
vector
:=
(1,0,0,
.
.
.),
is determined
by
the
equation
(/)Q and hence its
unique (up
,jc) representative Q(xi, phase factor) solution of .
.
.
0
-
,
in the
Schrödinger representation
to a
xj +
^)fi(x^,...,x)=0
.
ö^jj Thus
Q(.Ti,X2, and the
.
.
representatives
.
,X,0
-
of the
7l-"/2g^p^_^.^2 _^ .^2 _^
multiparticle
states
are
.
.
.
+;c,^)/2)
given by
,
is the
36
States in
*
Quantum Statistical Mechanics
(/;, )'
(f,.. )ß
=
2-"'/^
This Identification determines the
n
(-^A 4;) "^'" -
'
'
'""
^
equivalence of the Fock and Schrödinger
re-
presentation.
Physically in
n
the
is
one can
dimensions. The
^(ifj] correspond
given by
interpret
L- (R")
to
the space of states of
äs
^(/y) correspond
to the
position operators,
one-particle moving or observables, and
the coordinates of momentum. The Fock number operator
the differential operator
^-EMV^+^-i)/2 ./-l
and in the
Schrödinger picture this is interpreted äs the Hamiltonian of a point of vievv Q is the ground state, the state
harmonic
oscillator. Thus from this
of lowest energy, of the oscillator and the creation and annihilation operators create and annihilate quanta of energy.
EXAMPLE 5.2. 17.
Let
A(^']
denote the unique invariant a
scalar
product
on
be the almost
mean over
periodic functions over R" and let M Section 4.3.4). The mean M defines
y4([R") (see
^(R") by
(f,cj]=M(fg] and the
of ^(IR") with respect to the associated
completion norm is a Hubert space . dimensional, one can define a representation of the associated Weyl algebra on ^(R") by exactly the same construction given in Example 5.2.16. Extension by continuity then gives a representation on . This representation is not, however, regulär. For example, if / ^ 0 and O(.TI l, then .x,,] If
I) is
n
,
.
.
.
if ?:-0
(.^('/))
=
.o,
if./o
Note that the Hubert space is nonseparable, {exp{/ ^"^ forms an | V"^;}; ^i ^ ^} orthonornial basis of the space. A similar nonregular representation of the CCRs is given by Rf, in the proof of Theorem 5.2.8.
We complete this discussion of normal states with a few comments on a special subclass, the finite-density states. Theorem 5.2.14 establishes that co is normal if, and only if, D(naj) contains a cyclic vector. Biit there is a preferred cyclic vector Q.o associated with oj and we call co a finite-density state if
ü(o
G
D(n,o}-
to the Fock
Thus
co
has finite
density if,
representation and O^^
^
and
only if,
D(NcJ~}-
co
is normal with respect density matrix on
If P is the
Fock space which determines co, then the latter property is
positive selfadjoint operator N^^~pN^^~ having a finite define a positive functional, the number can A^(co) G [0, oo] by
N(co}
=
sup ^
if ^l is the CAR
algebra
and
Y. {/.}CF
eqiiivalent
to the
trace. Note also that
functional,
o,(a\f,]a(f,))
co G
one
E^y^
K^
Continuous
Quantum Systems.
37
I
Ä^(co)=sup ^ {t-^-o4(W(tfi)-^r(W(tf,}-^)) ''^
(fi}CF
+r^-co((W(itfi} for the
Weyl algebra,
and then
-
1)*(ff (/^/,-)
has finite
co
-
density if,
1)) and
-
l}/2
only if, N(o})
< +oc.
In both formulae the suprema are over finite orthonormal subsets {//} of l) and the assertion for the CCR algebra uses Observation l of the proof of
Theorem 5.2.14. Note also that in the CCR
that
N(a)}
< +00
case
it follows from the
^(^oX/))
that Q,o ^
implies
^^
7V(a;)-sup ^
for all
proof of Theorem / and
5.2.14
\\a,,(fi)Ü^\\~
{/J^F
analogy with the CAR case. Moreover, if A^(co) < +00, then its value can be interpreted äs the extension of co to the number operator and one has the in
identifications
7V(a;)
=
|Ky2^|P=Trg(7V'/VAfi/2)
.
It is of some interest to remark that A^ is affine, because the suprema are net limits, and lower semi-continuous in the weak* topology, because A^ is the upper envelope of a family of weak*-continuous functionals. In particular A^ A^, and respects barycentric decompositions, by Corollary 4.1.18 applied to the subsets of states with density below a fixed bound, i.e., the sets
{(JO',N(CD) Next
A},
<
by lower semicontinuity. properties of states with respect to the operators and, in particular, the determination of a
weak*-closed
are
examine
we
more
annihilation and creation
detailed
by its values on these operators. Discussion of states over the CAR algebra is simpler because the algebra contains the annihilation and creation operators (/), ß*(ör); f,g^^ and it follows easily that each co is determined by State
a)(a^(fi}-"a^(fn)a(fn+i)"'a(fn+ni}}^n,m>0,fiel).
the set of values
similar result is also true for If
co
is
/El)
and
regulär
a
state over
of the
^co(f}
generators
a
Weyl
A
regulär states over the Weyl algebra. the Weyl algebra ^(f)), then the infinitesimal
subset of
groups ^ G IR
i-^
T^o^(^(tf]]
are
defined for all
introduce the annihilation and creation operators a^^(f] and have mentioned above, the cyclic vector Q^Ü is not necessarily
one can
fl*^(/). But,
äs we
in the domain of these operators. Thus it is generally impossible to define analogues of the co(fl*(/i)a(/2)), etc. For this reason it is natural to introduce more
stringent
A State
oj
notions of regularity of states and their associated
over
^(t))
cychc
vectors.
t^o}(W(tf)) is m no)(W(tf))il/ is m times
is defined to be in the class C" if
/ G l). Similarly, \l/ G C" if strongly diiferentiable for all / G l). This latter condition is equivalent to lA G D(^fo(fr} for all /, and it is not difficult to see that co G C''" if, and only if, QOJ> ^ C"^- In particular the value of C^ states can be defined on all poly-
times differentiable for all
nomials of the
Oco(/),
CO(^co(fl}
and '
we use
^co(fn})
the natural notation =
(a., ^o.(/l)
^.o(/.)^o.)
,
38
States in
Quantum Statistical Mechanics
co(^a),,(/i)-.-a)^(/)) etc.
Finally
functions
C
a
^ G [R
state
^
co
co(W(tf}}
state
,
is called
analytic (entire analytic) whenever the analytic in an open neighborhood of the origin /El). There are two interesting properties of
are
(are entire analytic), for all analytic states. First,
(Qco,7r,(v4)a),,(/i)...(D^(/,Oao)
=
is
analytic if, and only if, Q^o is an analytic vector for all example, if Q^o is analytic for ^co(f}, then t\-^naj(W(tf))Q.fo is strongly analytic and hence t\-^cü(W(tf}) is analytic. Conversely, if co is ana lytic, then Q.aj is a C vector and
^(jo(f)-
a
co
For
(^(f/))-y^(ß,'(/)'Uo) l
>
.
0
Therefore
,eR^^(_i)^||$4/)"Qf (2) >o
is
analytic
and
äs
(2)!
>
2^^"~^'>(n
-
1)!^
one
has
Ef7;;^ll*(/)""lO'<
>ov(-l)' for all
1^1
<
^ and for
some
/o
0.
>
+
~
Equivalently
En-ll^(/)"^ll<+~ ^
>
0
for
^ < ^, i.e., QCÜ is analytic for
unitary
and hence
one
has estimates of the form
l(Q,7ü,,(i^(^/))0^(/)2^^ao)i
\(Ü^,n,,(W(tf)]^,,(ff'-^^^)\ This second remark has
<
<
i(Q.(/)'^'o.o)i
,
|(Q,,a),,(/)2'^Q,)p/'|(Q,,(D,,(/)^"-2Q,,)p/^
.
important consequence. Weyl algebra is determined by its values on the set of Weyl operators {^(/);/Gl)}. But if co is analytic, then o}(W(f}) is de termined by the derivatives of the function t\-^o}(W(tf)) at the origin. Thus each analytic state is determined by the set of matrix elements, or expectation values, {cD(^a}(fT)]f^^,n> l}, Alternatively co is determined by the set Each state
oj over
{co(ß^(/)''floj(/)'")}. be
completely
an
the
Hence
properties
reconstructed from
of the
knowledge
analytic
The conclusion of these observations is that
algebra
are
{co(a^(/i) analytic
determined
<(/,Oßa;(/;7+i)
states
are
analogous
by '
'
'
the
states can, in
principle,
of either of these sets.
analytic
set
a^(fn+n:}); n, m
of >
states
co over
multilinear
O,//
to states over the CAR
G
I)}.
In
algebra.
the
Weyl
functionals this
Both
sense are
de-
Continuous Quantum
termined with
a
by
clear
family of functionals which physical Interpretation. a
expressed
are
in terms of operators
It is often useful to consider other classes of multilinear
truncated functionals, whose values events. These functionals
physical
by recursion relations which combinatorically somewhat easier definitions corresponding to the CCRs.
determined
are
different for the CARs and CCRs and
are
for the latter. Thus Let 3 denote
we
an
functionals, the
related to correlations between
directly
are
39
I
Systems.
first describe the
are
arbitrary index set and F a function from the nonempty complex numbers. We associate with F the truncated
finite subsets of 3 to the
function FT by
the recursion relations
^(^)
En
=
^T(J]
,
^/ J&^I
where the
sum
Explicitly ^/
over all partitions ^/ of the finite set / into ordered subsets. {Ji J2, ..,/} where //n J/ 0 if / ^ 7, the union of the J/ is /,
is
=
,
and the elements of each // retain the order of /. For
F(a) and these
-
Fr(a)
=
=
Fr(a, ß]
+
FHa)Fr(jS)
one
has
etc.,
,
have the Solutions
equations
Fr(a)
F(a, ß]
,
example,
F(a)
Fr(a, ß)
,
F(a, ß)
=
-
F(a}F(ß)
etc.,
,
It is often useful to remark that
F(I)
^
=
FT(J}F(I\J)
aeJC/
for any / relations
3 a,
where
we
relations
take
F(0)
=
^ n
directly from the recursion arbitrary term FT(J) in these
l This follows .
that the coefficient of
by noting is given by
an
f^T(K}=F(i\j)
.
^/V ^^^i\J These
analytic
algorithms state
can now
co over
then the values
the
be used to associate truncated functionals with
Weyl algebra
^ü)(fn})
co(Oa)(/i)
on
of
function
defines
truncation COT which satisfies
over
COT(^M})
a;r(0,(/i),Oa;(/2)) By
linear
combination,
Let 3 consist of elements of
0; on
monomials of the
=
co((D,(/))
-
or
reapplication
of the
a;(a^(/i)
,
same
of the nontruncated functionals
determined
by
and,
linearity
'
one
,
etc.
also ob-
-(ö^w))-
and
^^ i^
antilinearity
moreover, that the state
co
is
the truncated functionals.
The truncated functional tween the
procedure,
a'^(fn)<^co(g\)
evident that the truncated functionals inherit all the
properties
a
procedure
a;(0,X/i)^a.(/2))-co(0,(/i))^(Oe.(/2))
tains truncations of the functions
an
1) and
define
^w(/)
the ordered finite subsets of 5. The above
complex a
1^.
operations
007(0^ (/),flaj(ö^)) represents
of annihilation of
a
particle
the correlation be
with wavefunction g and the
40
States in
creation of
Quantum Statistical Mechanics
particle with
function
/. The higher truncations represent demonstrate that the equilibrium States of a System of noninteracting bosons have the simple property that the higher-order correlations vanish identically and this property can be used to characterize a special class of states, the quasi-free states. Explicitly an analytie State co is called quasi-free if
multiple
a
wave
correlations. In Section 5.2.5
we
o;r(^.o(/i),...,a),,(/,))=0 for all two
2 and all
>
n
functionals
tionals must
/i
satisfy
be
.
.
certain
\\{^co(fi) can
.
,
fn
,
cor(^to(/))
reexpressed
+
Therefore, these conditions
quasi-free
coT(^(o(fi},^ü}(f2))a;($,,(/i)
-
+
In
state is
Of
determined by these func-
course
particular the conditions
/o,,(/2))naof
> 0
cor(0co(/2),a>.o(/2))-Im(/i,/2)>0
.
necessary for the truncated functionals to
are
state. What is less
a
a
äs
cor(a),,(/i),(D,,(/i)) determine
Thus
positivity properties.
i^co(f2}
+
f).
G
and
obvious is that this
set of conditions, for all /i,/2 G t), is also sufficient for the COT to determine a quasi-free state co, The sufficiency can be proved by explicit construction of the representation (co)^fo,^co) (see Notes and Remarks). In particular every analytie state 0) determines a quasi-free state which is obtained by setting COT(^OJ (fi)^
..,<^c.(/.))=0forall>2. EXAMPLE 5.2.18. space
9 G
l)
and
[0, 27r)
Let
assume
-^
that
^Q(W(f))
co
be
a
state
on
the CCR
algebra
over
the
pre-Hilbert
is invariant under the group of gauge transformations W(e^^^f). If co is analytie, then this invariance can be ex-
co
=
pressed by
<^(/l) if
n
^
m.
Hence
linear form
a
<(///)ßc.(ö'l) aco(g,n))
=
0
gauge-invariant quasi-free state is determined by a single sesquiand the positivity conditions reduce to the simple con
coT(a*(f],a(g))
ditions
^r(*(/),ö(/))>0 Conversely assume simplicity assume that pletion 5, of l), i.e.,
for all
f,g ^l).
the Fock
Let
vacuum.
.
that
one
it is
given by
a
t(gJ]
(T^''-g. r^/V)
,^
5^_(^)
=
If J is
an
has
0
=
positive sesquilinear form t positive selfadjoint operator
a
5+1)
and
-
a*^(g)
fl(vTTr/)
=
ß*(Vl
+
^ H + l 0
Tg)
T
on
t) and for the
com-
Q,^ -QoQ, where Q= (1,0,0,...) is
anti-linear involution
satisfying (Jf^Jg)
the operators
a,,(/)
over
a%J^/Tf)
(g) D + t O
a(jVfg)
,
=
(^,/),
then
Continuous Quantum
Systems.
I
4l
satisfy the CCRs on ,^. (The a and a* denote the Fock annihilation and creation (Qo^,^^) is a gauge-invariant operators.) It follows easily that the state (j^(A) quasi-free state and one has =
(:(/)<(
algebra
=
(J^f, J^iTg)
itself this state is
=
(VTg, vT/)
t(g,f)
=
.
given by
co(^(/))^exp{-a;((^J/)^)/2} -exp{-||/||V4}exp{-/(/,/)/2}
.
be used to reexpress mixing properties of a Bogoliubov transformations, in a particularly simple manner. This reformulation explicitly connects mixing properties and the lack of correlation between physical events. It is illustrated by the following example. The truncated functionals
state, with respect
can
to groups of
Let ^ denote the CCR algebra over L-([R'), or the quasi-local C*-subalgebra of 51 defined in Example 5.2.11. The group U^' of space translations acts äs *-automorphisms T of ^ via Bogoliubov transformations, e.g.,
EXAMPLE 5.2.19.
T,(^(/)) Note that the CCRs
=
W(U,f)
(U,f)(y)
,
f(y
=
-
X)
.
give
IIKv('(/)), W(g)]\\
=
|2sin(Im(C/,/,0)/2)|
<m.f,9)\ and hence lim
\\[r.(W(f)),W(g)]\\=0
.
|.X'|OO
It follows that ^ is
asymptotically
abelian for space translations in the
norm
sense,
i.e.,
||[T,(^),5]||=0
lim |.v|-^oo
for all A, B G ^. Thus if
co
lim |xHoo
for all yi,5
E
is any state
|co(^T,(5))
^, is equivalent lim l.vHoo
to the
co
-
^, the strong mixing property
a}(A)co('c^(B))\
three-body
|co(^T,(5)C)
for all ^,5, C G ^. If, however,
over
-
=
0
düster property
o}(AC)o}(i^(B))\
is R^' -invariant and
=
0
analytic
the
following
are
equivalent: is
co
(2)
lim|,|_,o ^r(^co(/i ),
for all ,
m
PROOF.
(1)
>
for space
strongly mixing
(1)
l and all
^
(2).
.
.
/i ,.../,
For g
.
,
translations,
^co(fn). Tv(a),,(gi )),..., T.(^c.(6'.0))
gf,
eL^(U'')
,...,
^,
and all
e
x
L^(!R^'), G
where
T,(O,,(/))
-
-
0
O(^x- /).
U'' the vector-valued functions
42
States in
Quantum Statistical Mechanics
te'a^n(w(tu,f))Q.^ have
an
of
This follows from the IR^'-invariance and
X.
analytic
extension to
strip
a
aroimd the real axis whose width is
power-series expansion and the density of the
eqiiivalent
(!') for all
to
lim|,|_^ (
(2')
independent analyticity of co. It then follows by Weyl operators that condition (1) is
L-Cff]
e
and all ,m
>
lim|,|_^ ft;r^(<&,,(/),
=
1. But
w((D(/)'>(
(T)
is
äquivalent
(D(/)^, T,((D(g)),.
/z-factors
for all case
/, ^
G
follows
Z-(ff^^')
and all 72,
m
to
T,('DXg))j
=
0
/-factors
> 1.
by inductive reasoning
This is obvious ^oi
^
F(/)=
n
=
m
=
l but the
general
from the formula
FT(J]F(I\J]
ae J c /
connecting a function and its truncations. Finally (2) and (2^ are eqiiivalent by linearity and anti-linearity. Note that the above properties all stem from the fact that 0 äs \x oo (L^x-/, 0^) and hence they generalize easily to other Bogoliubov transformations. Moreover the various results are valid if the pointwise limits are replaced by mean values. -^
-^
The notion of truncated function
can be extended to the CAR algebra ^ differences. First, there is a simplification because the a(f) and a* (g) are elements of the algebra and thus there is no inherent restriction on the States. The second difference is a slight complication of the combina-
I) but with
over
two
torics of the truncation process which is necessary for consistency with the anticommutation relations. This modiücation is only possible for even states, i.e., States which are invariant under the unique a of ^ for which
*-automorphism
(j(a(f)]
ß(/),/
=
Lei 3
ordered
once even
defined
now
I). The appropriate definitions
G
again denote
an
index set and F
subsets of 5 to the
complex
.^/
^i /
sum
{J\," ,Jn} (Ji 72, -,/)
=
K^
is
over
and
,
is
does not affect the
examples
follows.
recursively by
F(/)=^8(^;)J]Fr(J) where the
are äs
function from the nonempty numbers. The truncation FT of F is a
all
&(^i)
partitions ^i is +1
even or
even
or
or
,
JG.^/
-l
of / into ordered
according
odd. Note that
äs
the //
odd character of the
are even
their
permutation.
are
F(a,ß)=FT(,ß) F(a, ß,
y,
S)
=
Fr(a, ß, 7, .5) -
+
,
Fr(a, ß)FT(j, S)
Fr(a, y'JFriß, <5)
+
subsets, permutation
even
to whether the
Fr(a, d]Fr(ß, y)
interchange simplest
The
Continuous
and
by Inversion one obtains FT in terms of connecting the F and Fj,
Again
F,
I
43
simple
in-
Quantum Systems.
one
has
a
ductive relation
^
F(I)^
(J,/V)Fr(J)F(/V)
ae/C/
1 l and 8(J,/\J) according to whether permutation of /. If co is an even state over the CAR algebra, then the above process can be used to define a truncation COT äs a function over the monomials in the a*(f} and a(g). For example, one has
for any /3a where F(0) (JJ\J) is an even or odd
=
=
I\-^
CDT(a^(f),a^(g})^CD(a'-(f)a^(g)}
coT(a^f}.a(g)} a;r(a*(/i),*(/2),(^i),^(^2))
,
co(a^(f}a(g})
=
,
co(*(/iK(/2)a(^i)a(^2))
-
-co(a*(/i)a*(/2))co(a(^i)a(^2)) +
co(a*(/i)fl(öri))co(a*(/2)fl(ö^2))
-CD(a^(fi)a(g2])cD(a%f2)a(g,))
,
etc.
An
even
state
co over
^ is called
qiiasi-free if,
and
only if,
a;r(^(/i),a^(/2),...,a^(/.))-^0 for
and all
> 2
/7
Thus
/i
quasi-free
a
cor(ß*(/),fl*(öf))
,
.
.
.
,
/,j
E
a^ is used
represent either
to
two functionals
a"
or a.
coT(ci''(f).a(g))
by be given
and
can on any two functionals with the and anti-linearity conditions to ensure that they determine a
cor(a(f),a*(f}} which
where
and conditions
linearity quasi-free state co. Necessary
correct
I)
state is determined
+
correspond
and sufficient conditions
cor(a*(g),a(g))
+
(Or(a(f),a(g}}
are
+ ojr (a*
>
0
the Hubert space
l)
(ö), *{/))
to the conditions
||7r,,(a*(/)+a(^))Q,,||->0
EXAMPLE 5.2.20.
A state
co
over
the CAR
.
algebra
on
is
if it is invariant under the group of Bogoliubov transformations fl(e'^/),ö e [0,271} the so-called gauge transformations. A gauge-in
gauge-invariant
Tö(fl(/)) variant
=
quasi-free
,
state is determined
by
truncated function and the
one
positivity
conditions reduce to the conditions
11/11' >a;r(fl*(/),a(/))>0 The functional COT
automatically
determines
a
.
selfadjoint operator
T such that
l > r > 0 and
coT(a^(f).a(g)) If
cü
0.
is the state determined
=
by
co^, then the
?y_(WoS-(W,^w --Ü0Ü,
=
(g^Tf)
.
corresponding representation
where Q
=
(1,0,0,...)
is
given by
is the Fock vacuum, and
44
States in
In these last
is
an
Quantum Statistical Mechanics
a,X/)
-
dl(cj]
=
equations
r)^/-/) ry^'g} a*{(l a((^
äs
01+0
a(JT^^~g}
and a* denote the Fock space
a
satisfying (Jf.Jg)
=
,
.
representatives of the CARs,
(g,f\
=
and a* and satisfies 0Q
a
The tmncated functionals way
a^(JT^'"^f]
-
anti-linear involution
anti-commutes with the
same
1 + ö
-
and Q is
an
J
operator which
Q.
directly related to correlations in much the following example is the analogue of Example
are
for the CCRs. The
5.2.19.
EXAMPLE 5.2.21. translations acts
Let ^21 be the CAR
äs a
algebra over L-(R'). The group R^' of space strongly continuous group of ^-automorphism r of ^l whose
action is such that
T,(a(/)) The CARs
preclude
=
a(UJ]
that Ul is
(UJ](y]
,
f(y
=
-
x]
.
asymptotically abelian with respect to space IR"-invariant, then the pair (^, w) is [R''-abelian and co is automatically even. The proof of these Statements is essentially contained in the proof of Theorem 2.6.5. It proceeds äs follows. Let E^,) denote the projection on the subspace of t/o;([R^) -invariant vectors in ^,j and let ^l_ and Ul+ denote the odd and translations but if
even
co
an
is
elements of ^(. It follows from the CARs and lim Lr
if either ^
5 is
or
even.
monomial
a
approximation that
||[T,(/(),5]||=0
>co
Thus
[E,,7i,,(A]E,ß,,n,,(B]E,]
=
M,(E,,n,,([i:,(A),B])E,,)
=
0
by the mean ergodic theorem where M denotes however, A is odd, then a similar argument gives lim
an
invariant
niean
over
R''. If,
||{T,(^),^*}||^O
|.r|>cxC'
and hence
{En,(A)EE,n,_(A")E}=G Since
both
E,,Ti,,(A)E,
terms =
in
0. Thus
the co
is
anti-commutator even
In fact the [R^'-abelianness
and
(Ul,co)
coupled
.
are
with the CARs
M(co(Ai[T:(Bi),B2]A2)) for all [R^'-invariant states
co
and all
positive,
Ai.Bj
e
one
concludes
that
is [R'-abelian.
impHes
that
=0
*il. This follows
by first considering
even
B l and then
M(üj(Aii:(Bi)B2A2))
=
=
(Q,,, 7r,,(^i) 'o.7c,X^ 1^2/^2) a.)
M(co(AiB2i(Bi)A2)}
by the above asymptotic commutation for BI is odd and
A~,A^
denote the odd and
even even
elements and [R''-abelianness. But if
parts of /!/, then
Continuous
M(co(^ 1 1(^1)52^2))
I
Quantum Systems.
45
(^o,,n^(Bi)E,,n,,((A+ -^7)52^42)0))
-
-0
by the asymptotic commutation and anti-commutation, and E(^7i(o(Bi}Ej milarly for the second term of the commutator. Finally, one concludes that the following conditions are equivalent.
\imi^,\^^o)(Ai^(B)} o}(A)co(B) for all A,B e^. lim|,.|^oo cD(Ai,(B)C) a}(AC}a}(B) for all A, B, C G ^. ,5(/.), Tv(5(^i)), T,(5(^,))) lim,,,^^ cor(5(/i),
(1) (2) (3)
=
=
.
for all
Clearly (2) C
=
C~ + C^
the
.
-
=^
(1),
but the
are
the
decompositions
asymptotic lim |.v|>oo
.
/,,... /,^,,...,^,,where5(/)
lim {a)(AT,.(B)C) |.Y|cxD
by
0. Si-
-
co(ACi,,(B'^)}
Again
(D(At^(B]C)
(D(AC"^i^(B~)]
=
co(AC)o}(B'^)
+
Bogoliubov
+
=
o}(AC)cD(B+)
=
co
is
co(^C+)co(5~) oj(AC}cD(B)
even.
The
it should be remarked that the above
(Uxf.g)
by noting
0
a* (/))/2
that if B
of B and C into odd and -
=
,
=
even
.
B~ -i- B^ and
parts, then
co(^C~Tv(^~))}
-
commutation relations cited above. But then condition
where the last steps follow because äs in Example 5.2.19.
fact that
follows
converse
-
-((/)+
-^
0
äs
x\
-^
oo
and hence
-
0
(1) gives
a}(AC~)co(B~)
,
equivalence
of
(1) and (3) is proved
properties basically follow from the easily generalize to other groups of
transformations.
5.2.4. The Ideal Fermi Gas The
foregoing discussion of the CAR algebra, the CCR algebra, their states, representations was motivated by the theory of Systems of point particles. This algebraic apparatus provides a kinematical description of such Systems and it remains to specify the dynamics. The simplest thermodynamically interesting models describe noninteracting particles confined to and their
some
bounded open set A of [R\ the so-called ideal gases. Our immediate aim is equilibrium formalism for such Systems in the Gibbs grand
to describe the
canonical ensemble and in this subsection
we consider fermions. We begin with general discussion of noninteracting Systems and subsequently specialize to point particles in the configuration space. Let 5_(l)) be the anti-symmetric Fock space built over the one-particle Hubert space I). The dynamics of a System with an arbitrary number of fer mions moving independently is dictated by the Schrödinger equation a
in'^-^ on
5_(f)).
In
this
t), dT(H] denotes
dY(H)^,
// is a selfadjoint Hamiltonian operator on quantization of // discussed in Section 5.2.1, and lack of interaction between the particles is reflected
equation
the second
fi is Planck's constant. The
=
46
by
States in
the direct
Quantum Statistical Mechanics
siim
and tensor
product stmcture of dY(H], e.g., the w-particle energies of the n individual particles. We will choose
energy is the sum of the imits such that li l.
The solution of the
lA
5_(W^'Ar
G
=
exp{-f/Jr(//)}iA
and the evolution of any bounded g_(I}), is given by transposition äs A e
The action
the
on
the evolution
Schrödinger equation gives
^(5_(t)))
=
observable, i.e.,
any bounded
operator
on
,itH\ ^-itH\ T(e"")Ar(e ) AT^{
i,(A]
^
Y(e-^^^]^^J
=
annihilation, and creation, operators is particularly simple.
One finds
T,(a(/)) Thus, the evolution
-
a(e^^^f),
be
can
T,(a*(/))
=
a*(e'^^/)
.
expressed äs a one-parameter group of Bogoliubov algebra ^l(^). This group is strongly continuous
transformations of the CAR because
||T,(fl(/)) -a(/)i| Next let
=
\\a(e""f
consider the Gibbs
US
particle System K^^ denotes the
at inverse
~
\\e""f
=
canonical
grand
temperature jß
f)\\
e (R
f\\
-
.
equilibrium state potential /i
and chemical
of the G [R.
If
modified Hamiltonian
K^,
-
dT(H
then the Gibbs state is defined
-
MÜ)
=
dY(H]
-
i^iN
,
by
Tr(^-^^^M)
,^,
"(^)=ix^^^
'
the trace is over 5_(f)), and it is imphcitly assumed that ^~/^^/' Typically the individual particle energy is lower semibounded and this last restriction requires at least that ß > O.li can be reformulated äs a condition on the one-particle Hamiltonian. where^ G
'*ll(f)),
is of trace-class.
Proposition and let ^ G (1) (2)
5.2.22. [R. The
Let H be
follo\ving
a
exp{ jS//} is trace-class exp{-/? dY(H /(D)} is -
PROOF.
(1)
creasing)
order if
^
(2). /?
selfadjoint operator on are equivalent,
the Hubert space
on
l).
trace-class
on
5_(I)) for
all ^i G [R.
Let >
0
I)
conditions
{&n}n>(^ denote the eigenvalues of H in increasing (de(if jß <Ö) repeated according to multiplicity. If z e^^^', one =
has
Tr. (-'"''")=
z"'
^ 0
exp|-/j|^, ^
jO=l
Continuous
because of cess.
Quantum Systems.
and the definition of the second
anti-symmetrization
I
47
quantization
pro-
Thus
0
ET^^-^^'^""") /W
>0
J](l+ze-^^-)
=
W
>0
H exp{ze-^^'"} ;n
<
exp{z Tr(e-^^)}
-
m>Q
(2)
^
(1).
invariant
The operator
and,
Qxp{~ßK^}
leaves the
exp{-)5;^,}|t,-zexp{-^//}
The variable to
of
one-particle subspace f)
5_(I))
moreover,
z
which
occurs
in the
.
calculation is
foregoing
usually
referred
the activity. It often occurs in the sequel. The Gibbs equilibrium state is particularly easy to calculate with the äs
algo-
rithm
e-ß^^^a^(f]^za^(e-^^f)e-ß^^ This
algorithm
combined with the CARs
=
=
one
gives
liMi:!!^pM
,,,,.,,,,.
But then
.
zca(a(g)a\e-l^"f)) -zo,(a\e-l^"f)a(g])+z(g,e-l^"f)
.
has
<^(a^((\+ze-l<")f]a(g))=z(g,e-f"f) and hence
o,(a\f)a(g)) A similar calculation
(g.ze-ft"(l +ze-^^)-'/)
-
gives
-^^(n*(//)n
^(n*(//)n(^yO| v- 1
=
7-1
/
V'
=
2
)
j=\
:=x:(-o^-^-(^.,--^%)cofn^*(/.on(^^^^^) \i
p=\
=
2
j=l
J+P
-
zo;
L*(e-^%) n *(/.o n a(^,-)y v
Therefore, by linearity
and the
replacement
i
=
1
of /i
7=1
by (H
/
+ze~^^)~Vi
one
finds
48
States in
Quantum Statistical Mechanics
=E(-ir''^(^*(/iM^p))^^ n *(/') n ^(^v).
n^*(/')n^(^vo
^
V/=l
/
7=1
\i
P=\
2
=
7=1
/
J^P
Iteration of this
identity
expresses the value of
the
product of the *(//) two-point fimctions co(a*(fi)a(gj)). Note that if the number of ß*(//) and a(gj) differ, then the corresponding value is zero because Qxp{-ßK^^} leaves each of the subspaces {)'" invariant, i.e., co is gauge-invariant (see Example 5.2.20). Thus, the Gibbs state is a quasi-free state and
over
a(gi)
äs
the
of
sum
co on
of
products
^(f)).
Note that the above calculation is
a
direct
lution determined
the
by
generalized
Bogoliubov transformations
f,(fl(/)) In
particular
used the
we
=
to this
Hamiltonian
K^^, i.e.,
e'"^"a(/)e-''^"
=
e'"^^a(e""f)
5.2.23.
that
assume
=
co(^f,(a*(/)))U,^
Let H be
exp{ /?//}
grand
evo-
.
a
.
following.
selfadjoint operator
on
the Hubert space
l)
is trace-class. Let
o,(A)^ denote the Gibbs
to the
the group of
algorithm
The result of these calculations is summarized in the
and
chapter,
such that
o;(a*(/M)
Proposition
of the KMS condi-
application
tion, discussed in Chapter l and the introduction
canonical
Tr(g-/^^M) Tr(e-/^^/')
eqiiilibrium
state
over
the CAR
algebra ^(I))
and A G
the evohition
corresponding
where I.L ^ U. It folloivs that state is the
51(1)) h->f,(^) to
This
z
=
e^'^^M^-''^^^'
G
^(l))
generalized Hamiltonian Ki^^
dY(H]
i^iN
,
is the
iiniqiie i-KMS state, at the value ß, and that this gauge-invariant quasi-free state with t\vo-point function cü
co(a^(f}a(g)) where
the
=
e^^^^
=
(g,ze-l^"(i
+
ze"/''' )
'
'
/)
,
.
proposition
identifies the Gibbs
equilibrium states and the f-KMS discrepancy. The Gibbs states are only defined if exp{-jS//} is of trace-class or, equivalently, if expj-jßÄ^^J is of trace-class. These properties are unimportant, however, for the definition of the evolution f which exists whenever H is selfadjoint. Moreover, the notion of the KMS state can be directly defined in terms of f (this will be extensively discussed in Section 5.3) and the same combinatoric calculation given before the proposiStates
with
one
small
Continuous Quantum
Systems.
I
49
tion establishes that the
quasi-free state given in Proposition 5.2.23 is the Thus, the KMS states can exist when the Gibbs states are not defined. Nevertheless, one can establish that all the f-KMS states can be obtained äs weak*-limits of Gibbs states for dynamics f^'^^ which approximate f f-KMS state.
unique
in the
sense
established
that
||ff'\yl)
-
f^(^)||
by constructing //
-^
0
äs 77
for all A e
^(^). This result is (1) Qxp{-ßHfj} is of trace-class, exp{/W}. The groups T^"^ are then -^
oo
such that
^ (2) exp{/^//} converges strongly to defined by replacing H by H^ and the convergence on
and
states
of the
corresponding
Gibbs
follows from Lemma 5.2.25.
Now
examine the
thermodynamic limit of Gibbs or, more generally, noninteracting Fermi Systems. For the discussion we specialize to point particles in the configuration space U\ Thus we choose / eL~(A) where A is a bounded open subset of U^' and specify the one-particle Hamiltonian äs a selfadjoint extension of the Laplacian V". This latter operator will always be understood äs defined on the infinitely often differentiable functions C^(A) with support in A. The conventional quantum-mechanical Hamiltonian for free particles of mass m is fi^V-/2m but we now choose units we
KMS states, of
such
ihai'h^/2m
There
^
1.
many selfadjoint extensions of -V" on L^(A.), each corresponding to a choice of boundary conditions, i.e., a specification of the dynamical behavior of the particle on arrival at the boundary of the finite System. The number and nature of the possible extensions is partially governed by the smoothness properties of the boundary of A. We examine some specific cases below (see Example 5.2.26). If, however, A is replaced by IR^ then there is no ambiguity introduced by the boundary and -V^ has a unique selfadjoint ex tension H whose action is given by are
(Hf)(x) where
/denotes the
/GL^([R')
=
(2nr^^
J d^pf(p)i-e"'-^
Fourier transform of /. The domain of H is the set of
such that
|j>/|/(;.)P<+oo There
are
dynamics can
.
two basic
quantities of interest in the thermodynamic limit, the equilibrium states. For these latter one can take either the the KMS states. The following theorem describes the former
and the
Gibbs states but
,
or
easily
be reformulated for the KMS states.
Theorem 5.2.24.
For each bounded open set A c U^' let H\ denote a self adjoint extension, on L^(^], of the Laplacian -V" and let H denote the unique selfadjoint extension on L-([R^') o/-V^. Let ^A denote the CAR algebra over I^(A), M the CAR algebra over L^([R''), and r^ and T the groups of'-automorphisms of ^A and ^ such that i:^(a(f)] a(e^^^''^f) and it(a(f)) n that a(e^^^f). follows ^
=
States in
50
Statistical Mechanics
Quantum
(1)
lim^'^^\\i^'(A)
(2)
formlyfor infinite ö/[R, eventiially contains any given A C [R^'. If (DA is the qiiasi-free state given by
if(Ä)\\
-
0, for all
=
(ü^(a'-(f)a(g))
A G
(g, ze-^-(\
=
^U and all A
where A'
intervals
t
^
oo m
zß-/'^-^ )
+
'
'
the
/)
[R^
C
sense
imi-
that A'
,
then lim
coA'(^)
=
<j^(Ä)
A'->oo
fl// y4
/or and
G
^A and all A
is the
co
C
IR^', \vhere the limit
gaiige-invariant qiiasi-free
is in the
state over
sense
^ \vith
of (l)
two-point
function
co(a*(f)a(g))
(g, ze'f^ (\
=
^
[d^'p^)f(p)ze-l>''\l+ze-'>P'-r'
-
(27l)" f denotes
where
PROOF.
In
(1)
Example
+zg-''^)-'/)
the Foiirier
3.1.29
we
transform off.
showed that
||(/^'^-e'''^)i/.||
lim
-0
.
A->oo
The desired result then follows from the
continuity
\\^'^(a(f))-^,(a(f])\\ and the fact that 2lA is It follows from
(2) of
a
bounded
co(a*(f)a(g)) quasi-free
=
condition
\\(e""'^-e""]f\\
uniformly generated by the a(/). Proposition 5.2.23 that co^(a*(f)a(g))
function
of //A.
Thus
the
convergence
is the matrix element
of
co^(a*(f)a(g))
and hence the
state
o}
Lemma 5.2.25.
Hubert space
is
a
weak*-convergence of the quasi-free consequence of the following lemma.
Let U^ and U be
,
strongly
continuous
iinitary
\vith generators iH^ and iH. Assiime that the
to
states COA to the
groiips,
net
on
the
U.^ converges
strongly, i.e., lim a
for
all It
\jj ^ $), uniformly for follows that lim
II ([/,,, -[/,)iA||=0
)-cxD
t
infinite
intervals
II (/(//)- /(//Ml
o/
[R.
-0
a>oo
for
all
PROOF.
\jj
^
9),
and
for
all bounded continuous
If the Fourier transform
functions f on
/ of / is absolutely integrable,
[R
.
then the estimate
Continuous
ii(/(//,) which follows from
-
f(H)m
j dt\m \\(e""'
<
allows
spectral analysis,
f(H,)^
/(//)iA
^
to
-
e""m
I
5l
,
conclude that
.
in supremum norm, and hence the convergence
/ are dense in Co([R) all/e Co([R).
But these
one
Quantum Systems.
follows for
Next let
One has
g,n(t) Qy^p{t-/tn}. g.n(H,)^|J^g,n(H]^.^^M a
=
=
and hence for
0
>
n
-
>
/?.
But
one can
fg,
e
\\(gn,(H,]
<
\\g,(H.)^l> a
choose
n
-
Co(IR)
a
> y.
-
f(Hm
<
>
the
foregoing,
^
an 777
a
ß
\\g,n(H}ilf
/3,
<
and
f(H}g,(Hm
-
+
\\(!n,(H]^
-
n
such that -
a
<
n
<
/3
7 such that
||/|U/3
\\f(H,]g,(H,)^l>~f(H)g,n(Hm
+||/(//,)|| ||g,(/^a).A a
i/^ and, by
Therefore,
||/(//a)i/'
for all
g,(H)m
-
and hence there is
||/(//.)0,(^a)V for all
>
oo
\\gn,(H^)^
for all
g,n(H]\l/
V y. Since
ij/
and
-
lAII
e were
+
\\f(H]\\ \\g,(H)^
arbitrary, /(//a)
Theorem 5.2.24 demonstrates several
-
n
<
converges
||/|U
strongly
to
/(//).
of interest. First, the
dynamics strongly continuous onealgebra ^ over L~(W]. Second, the thermodynamic limit of the finite-volume equihbrium states is uniquely defined and independent of the choice of boundary conditions, i.e., there is a unique thermodynamic phase. This latter point is valid for either of the possible definitions of the finite-volume states, the conventional Gibbs definition, or the definition äs a KMS state.^ Finally the theorem gives an explicit identification of the thermodynamic equihbrium state of the Fermi gas äs a quasi-free state co and this allows a detailed analysis of equilibrium phepoints
of the infinite idealized Fermi gas are determined by a parameter group of *-automorpliisms of the CAR
nomena.
Let
us
examine
some
of the most relevant features.
First consider the
dynamics. The group T with generator -V-, or the related ^D, is usually referred to äs ih^free evolution, The group with generator -V^ Riemann-Lebesgue lemma implies the property -
'
-V- may have
selfadjoint extensions //A on L- (A) which are not lower semibounded when boundary conditions however, //A is lower semibounded, and the operator exp {-ßH^} has a finite trace for ^ > 0. (See Example 5.2.26.) Hence the Gibbs definition is restricted to classical boundary conditions and positive values of the temperature ß~^ The KMS definition is, however, valid for positive or negative temperatures and any boundary condition, and negative temperature states can be interpreted äs limits of Gibbs states for neighboring dynamics. This difference between the Gibbs description and the more general KMS condition could be of significance in more realistic models because certain phenomena of magnetism indicate the possibility of attaining negative temperature states (see Notes and Remarks). v
>
2. For all classical
.
52
States in
Quantum Statistical Mechanics
lim
(f,e""g}
/
lim
=
\t\rOG
d'pf(p)g(p)e"P-
=
0
.
|/|^00 J
Hence the
properties
derived for
an
of
IR-abelianness, asymptotic abelianness in mean, etc., Example 5.2.20 are applicable. In particular,
DR-invariant state in
M(cD(Ai[i(Bi),B2]A2))^Q for any [R-invariant state co, any invariant mean M, and all ^/,5/ G ^l. We will derive more detailed properties of the free evolution in Example 5.4.9.
Second, consider the equilibrium
state oj. This state is invariant imder the of time translations and under the group [R'' of space translations, i.e., the group of Bogoliubov transformations induced by the unitary transformations U; (U^f}(y} f(y x) of I-([R'). In fact the state is strongly mixing for
group
T
=
-
both groiips. This can easily be verified by application of the Riemann-Lebesgue lemma to the two-point function. The state CD also has finite particle
density
per unit volume p(ß,z) for 0 < ß < oo. Recall that in Section 5.2.3 we associated with each state
algebra
a
the state.
number functional
N((jo)
Thus, by restricting
number functionals
A^A(CO).
co
which
measures
to the local
Then
algebras ^A
N^(oj)/\/\\,
co over
the number of
where
|A|
one can
is the
the CAR
partides
in
define local
volume, i.e.,
Lebesgue measure, of A, is exactly the number of partides per unit volume in A. Since the equilibrium state is invariant under space translations, this number should be independent of A and indeed one easily computes the density äs
p(ß,z]
=
\^\-^N^(o,]
=
|A|-^^co(a*(AM/,)) /;>0
=
f d^p ze'I^P\l +ze-I^P-)-^
(27i)''
< +00
,
J
where
{fn}n>Q
5.2.14 that
presentation
co
is
an
is
orthonormal basis of Z.^(A). Thus it follows from Theorem
locally normal, i.e.,
to evaluate the local energy per
particle,
algebras.
It is also
re-
possible
unit volume, and hence the energy per unit an algorithm for the local
but for this it is first convenient to introduce
Hamiltonians Let
normal with respect to the Fock
in restriction to each of the local CAR
{/;/},;>o
on
be
Fock space. an orthonormal basis of 1-(A) formed of
differentiable functions and define
a
quadratic form
^A
once-continuously
on
the Fock space
5_(A) overL2(A) by
t^(l|^]
=
Y.(^|^^a\Vfn]'a(Vfn)l|J] n>0
with the domain of ^A consisting of those }// for which the sum is finite. It follows from Lemma 5.2.13 that ^A is a positive, closed quadratic form and an
explicit calculation shows that P_gi (^02^^ ---^ g^ ^ ^(^A) for all once-continuosly differentiable gi and all m> l. Thus /A is densely defined. Hence, there exists a positive selfadjoint operator TA on g_(A) such that
Continuous
?AW for all
il/
G
D(t^}. Choosing i/^
ticle vector with support in
I
53
(rl/V,7'l^V)
=
to be
A,
Quantum Systems.
an
infinitely-often
differentiable one-par-
calculates that
one
rAW--(^,vV) one-particle space, is a selfadjoint extension generally one deduces by calculation with multiparticle vectors that TA is a selfadjoint extension of the second quantization r(-V^) of V". In fact, it follows from the classical theory of quadratic forms and differential operators that TA corresponds to the second quantization of the selfadjoint extension of V" which satisfies Neumann boundary condition (see Example 5.2.26). This extension is very convenient for the calculation of the local energy per unit volume (j5,z) of the equilibrium state co. One has and hence TA, in restriction to the
of -V^
on
CQ(A).
More
(/?,z)
|Ar'^co(a*(y/)-a(y/))
=
>0
(271)-'
-
/ J> p^ze-f^P\\ +ze-f^P'^''
^(^ß)^^^^~^' ["^^'^ log(l+z^-/^^^) (The
equahty follows through integration by parts equation of state
last
classical
P(;ß, with the pressure
P(ß, z)
P(ß, z} The temperature
by
ß-\2nr
dependence
and expresses the
Z)=^(^, z)
identified
=
.
J
\^r /
J d^'p \og(l^ze-f^P"}
of p, e, and P is
p(ß, Z)-A-V(Z),
s(ß, z}
.
particularly simple. =
ß-^r''j(z)
For
example,
,
where /
/(z)-7C-^'/- [ d'xze-^'^l J(z) are
independent
7i-^'/'
f d'x zx^e-''\l
^-x"\-l ze
-t-
ze
(4nßY^^ corresponds ß and A particles, i.e., >! is a measure of
of
of the individual
=
+
=
to the thermal
wavelength
the "effective" size of the
fermions.
Quantum mechanically the Fourier variable äs
particle
momentum and the
p
conjugate
formulas for p and
(27i:)-'ze-^^'(l
-f
e
to
x
is
indicate that
ze-f^P'r^d''p
interpreted
54
States in
Quantum Statistical Mechanics
should be
interpreted äs the momentum distribution per imit high-temperature-low-density region, A^'/^p<^ l, one must have
p(ß, z)
~
(Inr^jd^P e-^"",
B(ß, z)
=
(InY'z
j d^'p i-e-^"''
Thus the momentum distribution takes the Maxwellian form and
by
use
of the
equation
of state
one can
P(ß, ß] p(ß, lO In
2s(ß,^) vp(ß,ti)
derive
=
fact, this is the ultimate justification of ß
suitable units. The strated
limit
ß
^
zero
(2n}~^'ze~f^P~d^'p
Boyle's law
the inverse temperature, in is best illuregion,
äs
}J^^p^l
temperature, i.e., by taking the
One then has
00.
lim
the idealization of
.
ß-
low-temperature-high-density
by examining
volume. In the z<^ l and
ze-f^P\l ^
+
ze-^P'r^
=
^
/j->00
^-^(^'-^'^(1
lim
+
^-^(/''-/O)-!
ß~^00
^ri, ifp^<^i l Thus
only particles with energy This Situation is often described
if
0,
p'^>^i
( momentum^) äs
the Fermi
sea.
.
less than
or
equal to /.i occur. particle energy
All states with
less than /i are occupied and all states with energy greater than /i are empty. The critical values of /?, the surface /?" /i, are called the Fermi siirface. One can also conclude that the weak*- limit of the equilibrium states äs =
ß
^
oo
function
exists and is the
gauge-invariant quasi-free
state COQ with
two-point
given by
(ao(a*(f)a(g))
=
(27i)-'' / '^
d^'p g(p)f(p] ^
P~
.
^ f^i
The zero-temperature states are usually called ground states and can be independently defined by the requirement of minimal energy per unit volume at
fixed
density.
The
general definition
Section 5.3. Note that the energy ground state of the Fermi gas and
g
of
and
one
behavior at low densities and
ground states will density p are easily
has
^
p^
^
^/^',
be examined in
calculated in the
in contrast to the linear
high temperatures. To complete the thermodynamic description one should also prove that the local density, energy, pressure, etc., are equal to the thermodynamic limit of their finite-volume counterparts. These limits are seemingly more sensitive, however, to the manner in which A oo and the boundary conditions vary. In the simplest situations one can, of course, calculate explicitly the limit and for the classical boundary conditions various techniques of convexity, monotonicity, subadditivity, etc., have been developed which apply even to interacting Systems. These methods are mostly based upon the characterization of selfadjoint extensions of V" by quadratic forms. -^
Continuous
Quantum Systems.
I
55
EXAMPLE 5.2.26
(The classical boundary conditions). Let D denote the Laplacian V^ defined on all twice-continuously differentiable functions in L~(A.)
operator
where A is assumed to have
a
piecewise
(D^,
a
differentiable
boundary ÖA. Green's formula
jdS^-j^^v-^^
=
restriction // of Z) to
a
domain
D(H]
is
Symmetrie
if
dif/
-dq)
^W^=W^'^ the
öA of A for all (p,
i/^ 6 D(H). Here we have used d/dn to denote the simplest and most basie examples of the boundary conditions are (1) Dirichlet conditions, \l/ 0 on d\. and (2) Neumann conditions, d\jj/dn 0 on dij/ where a e C^ (dA.} is a real differentiable function over dA (Di richlet conditions formally correspond to er +cx)). These conditions determine a family of selfadjoint extensions H^ of V" which are best described through quadratic forms. Let h^ denote the quadratic form defined by D(h^) C^(A) and on
boundary
inward normal derivative. The
=
=
=
=
hW This form is closure of
/z^,
=
IIY-Ail'
.
densely defined, positive, and closable (V is closable which we also denote by 7?^, determines a selfadjoint
on
C^(A)).
The
extension H^ of
-V2 by
h\i^) and this operator
corresponds
the restriction of
/z^
manner.
to
llx/^iAll'
=
to Neumann
CQ (A)
conditions. The closure h^ of
Moreover, the quadratic forms h"^ defined by D(h^)
h^(^^)=h\^) determine the there is
boundary
determines the Dirichlet extension H^ in the
d\jj
an a >
=
a^
=
D(h^]
same
and
j dSa\^f
+
extensions. These latter forms
are
closed because for Z?
<
l
0 such that
dS
a\\lf^^ <ö||^|P
+
bh\\lj)
.
Note that if 0 < 0-1 < 0-2, then
h^ in the
<
/z^'
<
/z^2 < h"^
of
denotes the
quadratic forms. It follows from the mini-max principle that if {ej;}>i eigenvalues of H" arranged in increasing order, repeated according to
multiplicity,
then
sense
pO <^n for all
77
<
p^\ ^n
<
0^1
^n
<
C.OC
^n
> 1.
If A is
a parallelepiped, -V" also has a selfadjoint extension //P^'" corresponding periodic boundary conditions. This extension is determined by the restriction 7?^^^ of h^ to the periodic functions in D(h^] and hence one also has
to
56
States in
Quantum Statistical Mechanics
h^ and the
corresponding order Finally let /z^, {e,7(^)}>i
Variation in A. If AI
C
A2
for the '
h^^^
<
<
h^
eigenvalues.
^^^'
one
has h'^_
>
eigenvalues
and consider
a
kf, and hence
C(Ai) >C(A2) for all
This property is often useful. For example, one is trace-class for all parallelepipeds and hence, by trace-class for all A. n.
exp{-//^}
can
compute that
monotonicity,
it is
We will not consider the limit of the
thermodynamic functions in any detail by noting that if A/, is a parallelepiped, with edges of ,1/v, the density with Dirichlet boundary conditions is given by
but Content ourselves
length Li,Z/2,
.
.
.
PA, (A z)
=
=
\\,\-^i:,(ze-l^"^^(\ + ze-t^"^..)-'} (L,L2---L,r' Y^ ze-^%('^)(l
where the
eigenvalues
are
ze-/^%(^))-'
+
,
,.--,'^'>l
ni
given by V^
i\\
f^^'^\
^
%(^)-2^(-^) V^^/ i=\
An identical value HI the local
expression
density
and hence
conditions. Thus the any
er
>
is valid for Neumann
0 is also allowed. But the
=
0
same
p^^(ß,z)
dxze-^''\\
+
is
boundary conditions but the a Riemann approximation to
just
tends to
p(ß,z)
for Dirichlet
conclusion is valid for the
by monotonicity.
/'
sum
Note that
d\lj
or
Neumann
ai// conditions
with
äs
ze~^^'Y^ >ze-^''\\
ze"^'^')
+
Jn-
//2+1 dxze-^^^(\
+
ze'^^T^
conclude that p^^ < p for the Dirichlet boundary condition and obtain upper bound on the difference p p^^. An explicit calculation gives
one can an
^^'
p(ß,z) where }^
=
(471^)^1/2 '^
:, is
-
p^^(ß,z)
the thermal
^
^^
<
l
+
wavelength.
Z
iHi;
One also
'
easily obtains
the
es-
timate
4 l
Combination of these one
<
P(ß^^^
-l- zA-
inequalities gives a bound replaces p^^ by its thermodynamic limit,
on
the
error
occurring
when
Continuous Quantum
-%^<12^ p(ß,z)
0<1
Systems.
57
I
.
L
Explicitly if L
=
3K,
l
one
cm
the
error
root of the
has /i
the
~
2
x
10~^
cm
for helium at
room
temperature and hence
is about 2 parts in 10^. At lower temperatures for example would be about 2 in 10^ because l is proportional to the inverse error
temperature, but for heavier
the infinite-volume Hmit
whose diameter is
äs
provides
small
äs
l
an
atoms the accuracy
excellent
improves. Hence, approximation even for Systems
cm.
5.2.5. The Ideal Böse Gas
begin the description of the ideal Böse gas with a general discussion which parallels that of the previous section. Subsequently we specialize to particles We
in r. Let 5+(f)) be the Symmetrie Fock space over the one-particle Hubert space I). The dynamics of the noninteracting System is again defined in terms of the second quantization dY(H] of the one-particle Hamiltonian H on 1^.
One has the evolution
'/'65+(W^'A. for the
wave
=
r(e-'"^M
=
T(e"")AT(e-""]
functions and A 6
for the observables. In
*-automorphisms
T
^(g^(!))) ^T,(^)
particular, the dynamics yield algebra ^(f)) and
a
one-parameter group of
of the CCR
T,(^(/)) for the
Weyl operators W(f]. Bogoliubov transformations is
=
W(e""f)
In contrast to the Fermi gas this group of strongly continuous because of Theorem
not
5.2.8. The Gibbs grand canonical equilibrium generalized Hamiltonian K^ dY(H /iH)
state is defined in terms of the
=
=
dY(H)
]jN whenever
exp{-^^,} is trace-class. This latter property of jn.
Proposition and let j3,/x
(1) (2) PROOF. Order if
5.2.27. G
R. The
places
a
Let H be
a
follomng
conditions
constraint
on
selfadjoint operator on are equivalent.
the
possible
values
the Hubert space
^
exp{-jS//} is trace-class on \) and ß(H yu1l)> 0, Qy.p{-ßdY(H A^H)} is trace-class on g+(f)). -
-
(1) ^ (2). Let {e} > Q denote the eigenvalues of // in increasing (decreasing) ^ > 0 (if ^ < 0) repeated according to multiplidty. If z e^^ one has =:
58
States in
Quantum Statistical Mechanics
exp|-/?^,
E
Tr(e-'''^")<^'
'^
I,rt2,...,,>o
p=\
and hence
0
Y,i:^^^(e~ß^>^)
=
m>0
0
[](!+ ze-l''^"(\
=
-
ze-"'^)'')
w>0
<exp|^ze-^^"'(l ^
-
ze~ß'-r'] J
ni>Q
<exp{z(l where the second relation
uses
ze-^'")"^Tr{e-^^)}
-
,
the
assumption ^(E^^ /^i) > 0 the to one-particle ÄT^^ space is
.
(2)
^
The restriction of
(1).
Qxp{ß(H
/.it)}
Identification
ß(H Let
Gibbs
-
/iH)
must be
of the
of
trace
-
/il
and hence
Qxp{-ßK^i}
that
jö(,
/.i)
-
>
0
for
all
m,
i.e.,
>0.
US now assume
equilibrium
that
exp{ jß^^J
is of trace-class and then calculate the
state
,^, "(^)
This is most
//
of trace-class. But then it follows from the above
Tr(e-^^>'A) =
^;:M^-
easily accomplished by extending co to the annihilation we first note that if i/^ G f)'^, one has
and
creation operators. For this
\\a(fi)a(f2)---a(fm<m"/^-m\\f,\\---\\f\\ Moreover
Qxp{-ßK/_i}l)''_^
C
1^'^
.
and hence
Tr,f.(e-l^^"/^-a*(f)
a(f)e-ß^"f^-)
a*(/i)a(/i)
<m"Trt,-(e-/'^")ll/iil'---||/f =
A
simple
(^^^"T^,l(e-^^'-}\\hf---\\f,.\\''
extension of the estimates iised in the
proof
.
of
Proposition
5.2.27
establishes that the operators
^^-a(/i)...a(/,)e-^^''/have
a
bounded closure
one can
extend
co
Äj-
and both
to monomials
or
Ä'j-Äf
and
polynomials
over, this extension is continuous in the
sense
ÄfÄ'^ in the
that
are
of trace-class. Thus
a(f)
and a*
(g}.
More
Continuous
Quantum Systems.
n
|co(a*(/0
.
.
.
a^(fn)a(gi)
-
a(g,)}\
<
a
suitable constant C. Now
one can use
e-ß^>-/2a*(f) in
to calculate
oj
brium state
prior
a manner
to
=
the
m
7=1
algorithm
a*(e-<^^"->^^^l^f)e-l^^'''^
similar to that used to evaluate the Fermi 5.2.23. In
Proposition
59
CjJ ll/dl H H^^'H /=!
for
I
equili-
particular,
(ü(a*(f]a(g))^^r(a*(e-l^^"->^^^l^f)e-l^^^a(e-f("-^^^l'-g))/1r(e-l^^"] a;(a(e-^('^-''^V2^)*(g-/((//-/'i)/2^)) f. l -ß(H(o(a (e -/J(//-^1)/2 f)^^^-ß(H-,mg)) + (g,e-ß("-^^'>f} =
=
,
where the last step
uses
the CCRs. Iteration of this
co(a*(f}a(g))
=
identity gives
^^^l^- f]a(e-"^("-^^^l^g)]
^(0,e-'^-"^)/V)
+
.
m=l
But
ß(H
;u1l)
-
>
0 and hence
||e-"/'(^-/'i)/2/||
lim
=
0
.
/2->00
Moreover, f^g\-^a}(a*(f)a(g)) is continuous by the previous observations. oo the above identity gives Therefore, in the limit that n -^
(o(a*(f)a(g))
=
(g, ze-l>"(^ -ze-^")-'/}
Repetition
of this method establishes that the value of CD
and fl*
determined
on
monomials of the
a
by sums of products of the above two-point function. Thus co is a gauge-invariant quasi-free state over the CCR algebra ^(I)). The value of co on the Weyl operators W(f) is then easily calculated. The quasi-free structure gives are
cü(ff (/)) and
one
exp{-ü;((l)(/)2)/2}
has
cD(^^(ff) The
=
=
w(fl(/)*(/) +a*(/)a(/))/2
=
(f,(^+ze-^")(^-ze-^"r'f}ß
foregoing calculation of co was based once again on the combinatoric provided by the KMS condition but expressed on the unbounded
relation
operators state
CD
a(f)
and
fl*(ö^).
It
can
be verified from the value of
also satisfies the KMS condition
taken in
phrasing the
necessary
the CCR
co(PF(/))
that the
algebra but care has to be continuity and analyticity properties because on
60
States in Quantum Statistical Mechanics
the evolution is not
of this The
strongly continuous.
We postpone the detailed discussion
to Section 5.3.
point following proposition
Proposition f) and let ß,
5.2.28.
summarizes the above resiilts.
Lei H be
a
^ G R. Assiime that
selfadjoint operator on the Hubert space is of trace-dass, ß(H j.ä) > 0,
Qxp{-ßH}
-
and let
"(-^) denote the Gibbs
grand
^^(I)),
dY(H
\vhere
K^,
It follows fiinction
-
that
CD
canoniccd -
Tr(e-^ Tr(e-/^^,.)
equilibriiim
state
the CCR
over
algebra
/iH).
is the
gauge-invariant qiiasi-free
(^(a'(f)a(g)) 077^
-
=
state
with two-point
(g, ze'^^ (\ -ze'/^^)-'/)
therefore
co(W(f)) Once
again
=
exp{-(/, (l +ze-ß^(^
remark that the
quasi-free
-
ze-^^]'^ f]/^}
.
occurring in the proposition eigenvalue for H, Example 5.2.18 shows that if ze"^^(1l ze~f^^)~^ is a positive selfadjoint operator, then the associated sesquilinear form determines a quasi-free state. It is not we
exists whenever //
>
/d jö ,
>
0, and
/:( is not
state a
discrete
-
necessary that the operator is bounded can
or
that it has discrete spectrum. One can be constructed äs weak*
demonstrate that this wider class of states
-limits of Gibbs states co corresponding to dynamics T^"^ which approximate the dynamics T defined by H in the sense that -^ o}(Äi:t(B}C) for
co,^(AI^f"\B)C)
all^,5,CG^(!)) and /G R. The infinitely extended ideal from the point of view of phase at low
Böse gas is
more
interesting
than the Fermi gas a phase transition
structure because it describes
temperatures. We first describe the properties of the thermodynamic
limit of the
equilibrium states and the dynamics in the single-phase region. Subsequently, we examine the details of the two-phase region. We adopt the notation used for point particles in Section 5.2.4.
Proposition V^
on
5.2.29. Let H^ denote
L~(A} corresponding
to a
a
selfadjoint extension of the Lapladan boiindary condition, and let H
dassical
denote the
uniqiie selfadjoint extension of-V^, on L^(U^'). Further let ^A algebra over I"(A) and ^ the CCR algebra over the siibspace l) ofL~(U'') formed by the iinion of the L- (A}. Finally let T^ be the groiip of -'-aiitomorphisms O/^A and ^, such that i^(W(f)) W(e^^^^f). Let ß > 0. denote the CCR
=
Iffollows that :
Continuous
(1)
Quantum Systems.
6l
I
If (o/^ is the Gibbs grand canonical state corresponding to (H\,ß,f.i) and if there is a C > 0 such that //A yul > CH for all A, then -
lim A.'
>
o:>f^(A]
CD(A]
oo
for all .4 G 9lA and all A C U\ The limit is in the sense that ^! eventually contains any given A C [R^' and CD is the gaiige-invariant quasi-free state over ^ vvith the t\vo-point function
co(/)a(6r))
(g,2e-ft"(l -ze'/''^)-'/)
=
l
^j d^'pg(p]f(p)ze-I^P\\ -ze-^n"' ~W) _
and hence
co(fF(/)) =exp{-(/,(1 +ze-f^")(^ (2)
If(>cü^
^w,
is the
^oj}
(a(^]" contains a representation ofthe \\(W^o(f]
(a)
all
f,
There exists
operators Uco
U^(t]^^ The Uca
G
and
l)
strongly continuous one-parameter ^^^ such that
=
Q
U^(t)W,,(f}U^(tr'
,
a
such that
a-weakly
x,(A]
=
=
continuous group
t/(/)^f/(r)~'
||/,
-
/||
-.
0.
of unitary
group
W,,(e'"'f)
.
of*-automorphisms
and
one
i
has
>00
lim A'^
(1)
||^||), for
+
on
-
for
C,\\f-g\\(\\f\\
co^,(^Tf'{5)C) =(Q,7r(^)T,(7r(5))7r(C)Q)
lim A
a
<
GZ2(r).
^vlim,_oo n^(W(fn)), iffn
=
implement
0/71(0(21)"
W,,(g]]W^(h]^,,,\\
-
g, h
W^(f}
(b)
PROOF.
.
cyclic representation corresponding to cü, then CCR algebra over Z."(1R^') by a family of unitary Weyl operators {W^^(f]\ f G L"(IR^')} such that n
(3)
_ze-/'^)-i/)/4)}
CD(Ai:^(B]C]
oo
all A, B, Ce"^. All the operators //A
Gibbs state is defined
only
condition //A
CH then
-
/xH
>
for
^
>
0
automatically lower semi-bounded and the by Example 5.2.26 and Proposition 5.2.28. The are
ensures
11 +
that
ze~^"^
0<:^^^^^<1coth(^C/2). Now it follows from
<^M(W(f])
Example 3.1.29,
converges to
co(W(f]]
Lemma
for all
/
G
^
5.2.25, and Proposition 5.2.28 that and hence coy converges to
weak*
topology. (2) Using the
CCRs
one can
successively estimate
that
co
in the
62
States in
\\n,.(W(f)
Quantum Statistical Mechanics
-
W(cj))n^(W(h))QM
|e-""(/-.")
<
-
1|
\\n,(W(f)
+
-
W(g))Q4
and
\\^co(W(f) But
W(g])^,\\
-
<
|e-M/,.)/2
_
l
+
\\n^(W(f -g)- i)Ü,,,\\
.
has
one
|^-,-In,(/-,,;,)
i|
_
<
|,-,M/.,)/2
||^_^|| ||;,||_
_
i|
<
||^_^|| 11^11/2
.
Moreover,
lk4i^(/-9)-i)a.iP
=
(i-(r(/-3)))
+
(i-co(r(0-/)))
<2co($4/-^)-/2)
.
The last step
uses the obvious bound on the two-point function following from the assumption ß(H /.fH ) > ßC"^ Collecting these estimates gives the desired continuity. Finally, äs QC,J is cyclic and H^cülö')!! ^ l for all g e l) one can define ^(7) for / e I-([R^') by continuity, i.e., if / e l) and /,, /, then W,^(f) is defined äs the .
-^
strong limit of
(b)
is
na,(W(fn}).
Define U
Waj(f]
and property
satisfied.
automatically
(3)
This establishes the existence of the
on
{W(f)ü^; f
6
L^(n^')} by
U^(t]W(f)a,
=
W,,(e""f]^,
.
It follows from the CCRs and the metric and extend to
explicit form of co(Pf((gf)) that the U,a are isounitary operators. The strong continuity follows from (2a), e.g.,
\\U^(t)W^(f]Sl,-W,(f]fl^\\<2C,\\e""f-f\\\\f\\ The invariance of
.
Q^o and the automorphism property follow by the definition of t/^oproperties with A, B, and C chosen to be Weyl
it suffices to establish the last
Finally,
(D^(Ai:^ (B)C]
operators. But then termined
by
can
of Lemma 5.2.25. A
use
identification. The second follows from the definition of
Remark.
T.
Statement
(1)
procedure
T^ and
one
be explicitly calculated and its limit destraightforward calculation gives the first the continuity of / e L^(U^')^^W(a(f] and
(3)
can
be extended.
By the
same
calculational
deduces that
limco^K^o<(^i)---<(A)) A
^
oo
=
(Qa
7r(^o)T,| (71(0(^1))
lim A'i,...,A' ^cx)
for all AI
(2)
It
G
03(A(,i'^^(A,)---x^:(An]]
^ and tt G R.
can
also be shown that
We will return to this
point
Under the restrictions
description
T((7tc(4,))^(o)
of the
co
satisfies
a
KMS condition with respect to
T.
in Section 5.3.
imposed the above proposition gives a satisfactory thermodynamic limit. This limit can be taken in a very
Continuous
Quantum Systems.
I
63
general form and the equilibrium state co is explicitly identified. Moreover, the dynamics is constructed in a natural manner äs a group of automorphisms of T^(D(^Y which appears äs a limit of the finite-volume dynamics. It is readily verifiable that co is invariant not only under time translations but also under the group [R^ of space translations. Moreover, it is strongly mixing for both these groups. Furthermore, co has finite density per unit volume and hence is a locally normal state over the quasi-local algebra generated by the ^A. The momentum distribution of the particles is now given by
(2nr"ze-f^^\l-ze-f^^'r^d^p and in the z<^
high-temperature-low-density region,
which
coincides with
again
l, this distribution is approximately Maxwelhan and agrees with the Fermi-
Dirac distribution. Let
next examine the
US
Since all the
boundary
shortcomings
selfadjoint
conditions
due to the conditions //
extensions //A of V-
corresponding
/il
lower semi-bounded, the conditions //
are
state that ^ < //o f^r some /IQ which
CH
>
.
to classical
/iH
>
CH
conditions used
the
boundary depends simplest case of dij/ a\l/ boundary conditions with a > 0 or 0 and hence ^ < 0 or z < L But an periodic boundary conditions, one has ^^0 of the calculation explicit density p(ß, z) gives on
for the //A- In the
=
j d^p ze-^P\\ -ze-^P~]
p(ß, z]
=
(27c)"'
\A
where A
integral
=
(^Tiß)^^'^
is finite and
is
once
one
;C
'
again
d\
IWAt^
-
1^1
fcT^
wavelength. If v > 3 the density at fixed temperature
last
the thermal
concludes that the
is
a
bounded function of the
activity. (The same conclusion is valid for all v if this But boundedness does not faithfully reüect the properties of large < 0.) ^0 but finite Systems. The density of a finite System A is given from Proposition 5.2.28 by
PA(^, z]
^
|Ar^Tr(z^-^^A(l -z^-^^^)'^)
-|Ar^^ze-^^"(^)(l-z^-^^^(^))-^
,
>o
where
e,j(A)
be made
are the eigenvalues of //A. arbitrarily large by choosing
first term in the above
sum
contributes
drawback of
Proposition
phenomenon
which is known
Thus for
ß
and A
fixed, the density
z
dose to
a
significant proportion
exp{^eo(A)}.
In this to
may
the
case
p/^(ß, z).
The
5.2.29 is that it does not take account of this latter äs
Bose-Einstein condensation.
In Order to understand the basic effect of Bose-Einstein condensation
on
the
of the Böse gas we examine the thermodynamic limit of the Gibbs States at fixed density, but variable activity. This discussion demands
high-density regime more
of all
detailed
analysis than was previously necessary. A complete description possibihties, e.g., all possible boundary conditions, etc., would lead us
64
States in
too far
Quantum Statistical Mechanics
astray and hence and
examine the
we
simplest
case, Dirichlet
boundary
conditions, mostly parallelepipeds A/, with edges of length LI, LI, jl'v Thus the local density p/<^^ is given by .
.
consider
.
p^^(ß,z)
=
(L,L,.--L..r'
E ^e-ß^^^^\l-^e-ß'^^^^r z
y
n\ ,...,/Zv > l
where 2 X
.
nni^ ^^r-^i nni^
X
%(^)=L(T", V^^
^
i=i
A Riemann
bound that
approximation argument establishes frequently use, and
lim AI^OO
for all
ß
>
that
p/^^(ß, z)
<
p(ß, z),
a
we
p^X/J, z)
=
^
0 and all 0 <
fd^'pze-'>P\\-ze-ß^T'
(2nr
P(ß, ^)
=
J z
l
<
.
but then
In
the
particular
l or v p(ß,l) is infinite if v zi->p(^, z) are both strictly increasing. Now =
limiting
value
2. Note that
=
z
l is allowed
=
z^-^p^^(ß,z)
and
begin the examination of the thermodynamic limit at fixed density by considering the Variation of the activity. The following result is crucial for the understanding of Bose-Einstein condensation of thermodynamic Systems.
Theorem 5.2.30. Assume
Gibbs
p(ß^ z) each p
State
the >
we
> 3. Let p\^^(ß^ z) be the par fiele density of the boundary conditions for a parallelep iped A^ and thermodynamic limit of pj^^(ß^ z). Define Pc(ß] p(ß-, 1) and for choose zi äs the unique root of v
with Dirichlet
=
0
P\,(ß^^L)
=P
-
Itfollows that:
(1)
If~p
Pc(ß]
<
and
z
is that
unique
lim A
where
(2)
\i
^
If^> pc(ß]
oo
z^
=
then
lim^^^^zi
äs
\L
^
of p(ß, z) =~p
z
.
,
.
.
Ly
,
l and
=
-^
if
oo.
(supi<^.<,Ly)/(infi
then
oo
|Air'zie~'''^^^'(l-ZLe-^'='^'^)''=p-pc(^)
lim
then
OO
indicates that Zi
remains bounded
AI
>
root
,
^00
where
O(AL)
of the
Dirichlet Hamiltonian
=
ei(A),
with l
=
H\^
(l, .
.
.
.
,
1),
/^ the smallest
eigenvalue
Continuous Quantum
PROOF.
For
The
proof is based
convexity and growth properties convexity of z\-^ p^(ß, z) implies that
upon
if Z] > Z2, then
example,
^P^fß
^
^(ß. z,] .
Systems.
P^(ß^ ^i)
^A(^' ^2)
-
""^"- ^'^
/
<
az
^-^
""^"-
Z]
^PA
zi) ^(^, öz
^
<
Z2
of the
I
65
density.
,
.
But
^(^,z) where
(A)
the
are
|A|-' ^e-^^.(^)(l
=
of //A-
eigenvalues
-
e-^^.(^);
_2
Therefore,
^A(^'^) ~z(l -zg-^^(^))
PA(Az)<9pA(M<. öz
~
Z
Combining
these
inequalities gives
P^(ß. ^2)
PA(jg,^i)-pA(^,^2)
^ ~
Z2
^1-^2
These bounds will be used As
(1)
PA(^, ^i) ~Zi(l -Zie-^^o(A)) ^
the remainder of the
throughout
ä^d t)oth functions
^ P()^5 -^) choosing z\
pAi(^5 ^)
ZL > ^- Thus
'
ZL and Z2
=
=z
proof.
increasing in z one must inequality one finds
are
have
in the above
o<.-.
ziz.
Ai-^oo
Assume
(2) one
has
a
p,(i?)
p. If z^ < l then
<
contradiction. Hence z/,
Next define
p^ and
p)^
p,(^)
PA,(^, Zi)
<
1. But Z < e^so(Ai)
>
g^
p(/?, ^i)
<
HmA^^oc-Zi
=
<
p,(/?)
and
1.
by
^,.j-'.,.-''^(^^)n -.,.-/'^(Mr' p^-limsup|Ai|~'zie-^'o^^^)(l -zie-^'^^^y -
^
A/,
^co
and
p
=
|A,|-' X:-e-''^='^''(l -ze-''^=('^'))-' /7>7M
where n>
m
indicates
/
> m/
for at least
PAL ^y omitting a finite number convexity properties of the p^^
one
l
/
of terms and the
,
.
.
.
,
v.
pj^^
Thus
p^
retain the
is obtained from
monotonicity
and
.
Since ZL
>
l
one
has
p^^l(ß, ZL] äs
the i
there is
(A/,)
term is the
p^^^(ß, ZL) p^^^(ß, >
p^^l(ß. 1)
only
one
A
>
|Az,||(Az,)
>
7c^/l~^|A/^|/(infi<^-
such that
PA,(^, 1)
omitted in the
a
l
>
(supj
cx)
<
Hence
1).
>
-
(^|A,|i(A,))-^
suni
for
m
/(infj
äs
A^
-^
oo.
=
]_. But, by assumption cxo. Therefore A^
äs
-^
Consequently
66
States in
pM
Quantum Statistical Mechanics
<\\mmfp^^^(ß,ZL} A >
''
00
liminf(p-|Ai|-'zie-''-^i('^'-)(l ZLe-^^^'^^--^)-^]
=
=p-p^
-
A/,
because _i (Az,) Next define PQ =
-^
\
00
/
O(AL)-
Hence
pj^
< p
p,.(ß)
-
.
by
Po
|AL|"^Ze-^'^o(^^)(l -z^e^^^o^^^^))"'
liminf
--
AZ.
^
.
OC'
It suffices to prove po > p Pc(ß) because this < e^^''^^^^^^ and consequently -
that 'p^
implies
=
p^ =^
-
p^(ß).
But
zL
|A,-W-^^=^(^^^)(I-Z,^-^M^^))-' |A^r'e~^(''^^^^^~'i^^'^^^Vi -e~^^^'-^^^^-^~'i(^''''^)h"'
<
<(^|A,|MA,)-i(A,)))-^ Moreover
|Az^|((A)
8j_(AL))
-
>
A^,
äs
oo
-^
for
oo
.
n^
L Thus
concludes
one
that
|Azr'z,e-/MA.)(i _z,^-/^^^(^^))-'
lim
=0
A/, ^00
for
7^ l, i.e., only density. Therefore Po
=
the state of lowest energy
limjnf f p
-
p)^^^ (/?, zi) j
provides
=
p
a non-zero
lim sup
-
contribution to the
p)^^ (j5, z^
A/, ^00
for all
m
>
i. The
p^
have, however, convexity properties similar
the p^^ and
to
hence
^(-)/. PA, IP' ^L) ^
_
where use
,(A/,)
the
_
>)/. lj PA, IP, |^
denotes the smallest
<
(--I-l)p!vf(^.^l) ^^^^
_
^^g-/je,(AO)
eigenvalue occurring
homogeneity assumption, (sup|
<
/
x-2
in the series for
}.(mfi<j<^,Lj)
äs
p;^\ Now A^
^
we
to
oo,
estimate that \-2
/
fii^(Az.) Hence
<
Tü^v
inf V<J<^'
2j_(A)/,(yV)
72g-Av.(A/j
^
Lj]
< l
<7i-v;r
Vi<7
/
for all
g-/j(c,(Ai)-2i;^(Ai))
sup
^ |
m
>
^^^
(2v)'/~/.
^^.^^
L.-
<
(v/lV/?r),(Ai)
Then for this ränge of
^^^ rearrange the
But i < Zi <
e'''-i''^''
<
p^^(ß, \)z,(l
and
one
-
z,e-l^^-^^'-))(\ -z^e-/'="('^'))-'
concludes that
m
one
has
convexity inequality
obtain
pif (^, z,)
.
/
.
to
Continuous
^-1 _
limsuppif (/?, zi)
<
ij.j.1
<
.:
(l
-
large
'
-^00
.
m
limsuppif (/J, z,) \l
and
r
2s,(Ai)/(Ai))
7-
_
p,(^)limsup AjT
^-ßm(^L}\
^
A,^oo
^00
67
I
% p, (^Z'-e-''^"''"')
p,(/J)limsup owp
-
A.
Quantum Systems.
pM(^
<
-
2v)?/m^)-^
>00
consequently
Po>p-Pc(/5)(l-2vAV''r' As the left band side is the limit
m
^
of w
independent
concludes that PQ
one
Theorem 5.2.30 is formulated for Dirichlet
Although
similar result is true for Neumann conditions
dij/
with
(j\l/
Y^]=,i(nni/Lf)^,
(7
0
>
boundary conditions
/
=
.
Pf,(ß, z)
|A|~^z(l
p^(ß,z)
The
.
,
|Ar'z(l
-
z)-'
+
+
all,
or
/
=
0.
r^(ß, z)
is the contribution from the zero-energy ground state, density and the remainder r\(ß,z] is the contribution
eigenvalues
with some, but not all, HI
v-l
r^(ß, z)
<
=
0.
Consequently,
TA
satisfies
^c,(L}(2nr^' J =
\
product n)=i(l +^/^-)-
is the coefficient of ;c^ in the
where
c^(L]
such
that
p
(p l AI
|)~\
< l
.
/ d^^p ze-ßP\l -ze-^^'r'
^
;U
p^(ß, z)
some,
estimate 0 <
TA
%(A)
spectrum
of the Neumann Hamiltonian contains the Di
is the Dirichlet
from the an
z)~^
-
=
.
a
the intermediate conditions
5.2.26).
Example
(see
or
0, l, richlet Spectrum together with additional modes for which Hence the Neumann density p^ has a decomposition
where
Pc(ß] by taking
> p
co.
pi^^(ß,zi).
=
Then
zi(\
suffice to deduce that =
-zi)~^
These bounds in combination with the
foregoing
0. (Gare has limA^^oo ^/^L(ß^ ^L) in the bounds äs integrals diverge äs z control the divergence.) Therefore
l , 2 terms
zi suffices to
=
the
p
>
<
Now fix ZL
and
\\mM\^L-\(\-ZL]-' ^p,(ß)
hence
zi
bounds
on
to be taken with the
l
.
But the bound
on
.
A/, -^00
Hence if p >
Pc(ß]
L/. Thus 0
l -z/^ <
<
zi(\ zi)~^ > (p ((p p^(ß]]\!^i\)~^
then
|Air%(l-zi)-^
lim
If, however, argument
p <
lim
=p-
Ai-^oo
p^(ß]]\/^i\ for limA^->oo^L
and
all =
sufficiently large
l-
Moreover,
P^,(ß,ZL)^p-p^(ß]
.
AL^OC
Pc(ß)
establishes
p(ß, z] that
then
< 1.
z
p/^^(ß, z)
>
But
p(ß, z)
a
Riemann and
approximation limA^-^oc PA^!^? ^)
68
=
States in
p(ß, z)
proof
=
Quantum Statistical Mechanics
p. Therefore ZL
of Thoerem 5.2.30
convexity argument
Q
iised in the
that
implies
.
limAi->oo^L ^- Finally, the spectrum of the Hamiltonian correspondboundary conditions d4f a\jj with 0- > 0 is sandwiched between the Dirichlet and Neumann spectra (see Example 5.2.26). Thus the density at fixed activity lies between the Dirichlet and Neumann densities. Therefore the activity z^ corresponding to a fixed density p is intermediate to the activities zi and zi for Dirichlet and Neumann conditions. Consequently z^ converges to the common limit of z/, and z/^ äs A^ oo. The thermodynamic limit is to this extent insensitive to the boundary conditions. The discussion of the thermodynamic limit of the Gibbs states at fixed density follows a very similar pattern. It is convenient, however, to first examine the limit at the critical value of z, namely, z l with variable density. This is particularly easy for Dirichlet boundary conditions and the result is shape independent. ing
=
to
=
-^
=
Proposition 5.2.31. Let corresponding to //A, ß^
L^(A),
and
"sHA
denote the Gibbs
CD\
^nd
is the CCR
z
grand
canonical
algebra
"iKA, 6>/ V~, on
s täte, over
\vhere //A is the Dirichlet extension
L^(A).
over
It follows that the \veak*-limits
co(A)
lim
=
co^'(A)
A'>oo
l and each ^ > 0, ^ G UA ^A "^^hen A^ existfor z oo in the sense that A^ eventually contains any A C [R^ The limit state CD is the gaiige-invariant qiiasifree state such that ^
(ff (/))
=
exp
{
-
(/, (1
+
e-l^")/(^
-
e-l^")f)/4]
*exp|-(27r)-^' l d^p\f(p)\^e-I^P-(\-e-ßP^)-^\
^g-ii/r/4,
J
\.
for
all
f
G
\^^L^(^),
L~(W). In particular, a}(W(f)) (D(W(f]] > 0 in all other cases. PROOF. erators
The
proof relies
exp{ JÖ^/A}?
property
)
\vhere H is the unique
ß
but rather
surprising property of the opChapter 6; see Corollary 6.3.13. This
derive in
we
states that for each
selfadjoint extension of V~ on if v=l,2 and j'd^xf(x)^Q, and
Q
simple
a
upon
which
=
>
0 and
/
G
^"(A)
the function
A'^(/,exp{-^//y}/) is
increasing
and lim A'
These results
->
(f,e-f^"^'f]
00
rely heavily
=
sup(/,e-^^A'/)
=
(f.e-^'^f]
A'
on
the choice of Dirichlet
boundary
accept them, then the proof of the proposition is simple.
conditions. If
we
Continuous
One has from
=
69
I
5.2.28 that
Proposition
o^^(W(n}
Quantum Systems.
e-\\f^-l^ exp
{ (/, e-'^'V)} -
*<
^
=1
and therefore
lim
cov()^(/))
-
ß-ll-^ll'/^exp (
-
l
A' -^00
=
supsupV2^'(/,e~"^^^'/)lJ A'
z
e-ll/ll'/^exp{-(/,^-^^(l
^g-ll/llV4exp|-(27r)-^ But for
/
e
,fr;
L^(A),/(p)
/ 0,
is, however, finite if /(O)
the
-
e-^^r'f}}
ld^p\f(p]\^e-^P\\-e-^P-r'\
-
l or 2. It integral in the exponent diverges for v /is analytic, and it is finite for general / when =
0 since
=
v<3.
Now
determine the
we can
brium states at fixed
thermodynamic behavior
for the Gibbs
equili-
density.
Theorem 5.2.32. Let COA^ t>e the Gibbs grand canonical state corresponding to {jc G !R^'; Li/2 < xi < Z///2}, where //A, ß and zi for the parallelepiped A^ V^ 077 Lr(^i) andzi is chosen such that o}\^ //Ai ^s the Dir fehle t extension o/ has particle density p. =
It
follo\vs that the weak^-limits lim
CD(A]
a)Ai(^)
AL-^OO
exist for each
Zj
,
.
.
.
,
LV
-^
ß oo
>
0,
cxo U A ^A' "^here A/^ remains (supi < / < yL/)/(infi / ^Li)
p >
and
0, and A
G
>
<
<
in the
sense
that
\vhen
bounded
Pc(ß)- ^^^ ^ ^^^ gauge-invariant quasi-free states. Let p(ß^ z) denote the thermodynamic density and set
P ^
a;(fF(/)) The
two-point function Paj(f^f)
(A)
=
exp{-p(/,/)/4}
^^ then determined
follows:
Ifp
P^(fJ)
(/,0 +ze-/*^)(1
=
where H is the
(B)
äs
.
unique
root
If p
p,(ß)
>
selfadjoint extension of of p p(jS, z).
V^
on
,
L^([R^')
and
z
is the
=
then
Pa,(/,/)-2^+Vo(^) vhere
-ze-f^")-'f)
ld^xf(x
pQ(ß)=p-p,(ß}.
+
(/,(l+e-^^)(1-e-^V/)
,
70
States in
PROOF.
Quantum Statistical Mechanics
(A)
Let p <
Pc(ß]-
there must exist Lf and 0
z
< l
As pp^^ (ß, z) < p(ß, z) one must have z < z^. such that z^ < z for all L/ > Z//. Therefore,
Moreover,
(/,(1] -^ze-I^^^L)(^
<
-ze-^^^LY^f) (/,(t ^z,e-ß"^^](^ -z,e-ß"-'r'f]
-
<
2||/|P(1 -z)-^(z, -z)
.
The convergence of COA^ to the correct state then follows from Proposition 5.2.28 and Theorem 5.2.30. The case p Pc(ß) will be covered by the proof of (B). (B) Let /;(A) and '(A^) denote the eigenvalues and eigenprojectors of //A/,- The State coAi is quasi-free and cD\^(W(f]] expj-p^^^^^ (/, /)/4} where =
=
PoM,(/-/)
=
EO +^Le-^'"-^^^^)(\ -Zie-'''=(^'))-'(/,(Ai)/)
.
n_
Now the first term in this
AI
^
00.
is
a
product
of two factors each of which has
|AJ-^(1 +zie-^'^^^^))(l -Ze-^^^o^^^))->
lim A/.
by
sum
a
limit
äs
First, -
2(p-pM}
-^00
Theorem 5.2.30; and second,
lim
|A,|(/,^o(Ai)/)-
AI^OO
=
where
lowest
we
lim A,
2^'
r
J
l
/
./
-.00
But for the
eigenfunction corresponding
to the
is
2"/2
MX)
f^
TT cos y \Li
^ V(^)
have used the fact that the normalized
eigenvalue
JV W
A
Jv/2
IA
/T: /
^l-,--y lAi TTäH^i^fG
=
higher
.
|Ai
.
n/nxi\ ._,^
terms
/rf".v|/W|
|Ai|(/,(Ai)/)<2" and lim A/^
l AI
-^(1 +zie-^^%(^^'))(l
-
ZLC-^'"'^^^^^]-^
=
0
*co
The first Statement follows
by explicit calculation and the second by the estimates used to prove the second half of Theorem 5.2.30. (This again reflects the fact that only the lowest energy state gives a nonzero contribution to the density.) Thus for each m > l one has lim
(p.,, (/,/)-P!S
.L-^CG
L
(/,/)) L
=
2"+'(p-p,(/?))
/ rfVW /
where
PJSl(/'/)
=
Y.(^+ZLe-^''-^'''-^)(l-z,e-f^--^^'-^r\f,E(\L)f)
Continuous
Finally,
lim
lim ->
7l
I
may prove that
one
/72
Quantum Systems.
p
(/,/)
(/, (l +e-l>")(^
-
-
e-"'')-'/)}
by repetition of the argument used to prove the second part one uses monotonicity of p)^ in z to obtain an estimate
P^^l(f^f} and, second,
>
-e-^^^O"'/)
{/,(1 +e-'"'^0(l
one uses
p^;;,(f,f}
<
in
convexity
to
z
(/,(i +e-^^v)(i
-
0
completed
of Theorem 5.2.30. First,
ll/f ^(^|AL|6,,(Ai))-'
find
-e-/'^^o-'/)i; "T-feJtÜ ([
The argument is then
=
^L
\i^00
00
zie
-
)
in Theorem 5.2.30.
äs
are two distinct regimes for the Böse high-temperature-low-density regime described by Pro position 5.2.29 and corresponding to z < l for Dirichlet boundary conditions. l and Second, there is a low-temperature-high-density regime in which z Bose-Einstein condensation takes place, i.e., a finite proportion of the particles
Theorem 5.2.32 establishes that there
gas.
First, there is
a
==
occupy the lowest energy state. In this second regime the System has a multitude of possible equilibrium states each of which has the same temperature and
The
activity. P ^
states
in the
by their particle densities and all densities two-point functions of the equilibrium states co are given by
differ
The
possible. condensation region
[Pc(ß}^ ^
^^^
=
+
and the local
y d^x^) J d^'xf(x} (2nr ld''p^)f(p)e-l^''\l-e-I^PT'
2Vo(^)
density by
p(ß) where
p{^(ß]
density
p
Pc(ß]
rp,(ß}
=
and p G
+
p,(ß)
[pc(ß},oo).
,
The term
po(Ä
and the coefficient 2^ indicates the
nieasures
particles in the lowest, zero, energy relative proportion of the condensate
of the condensate, i.e., the
the
state, at the
origin. This coefficient can be identified from our estimates äs the thermodynamic limit of [A^l |i/^o(^)l^ where I/^Q is the lowest-energy eigenfunction of the Dirichlet Hamiltonian H\^ If one replaces Dirichlet boundary conditions aij/, then a slight modification of the foreby the elastic conditions d^/dn going arguments establishes the existence of the corresponding equilibrium state. The only noticeable difference is that the coefficient 2^' is replaced by a positive constant c^, .
=
c,
=
lim A/,
where
I/^Q'
is
now
the
\M\K''(^}\'
.
-^00
lowest-energy eigenfunction with
the
new
boundary
con
ditions. One can check that c^ is a decreasing function of er and c^ ^ l äs er ^ 0, l for Neumann conditions. In the Neumann case the condensate is i.e., Cff =
72
States in Quantum Statistical Mechanics
distributed
evenly throughout the System and tlie local density takes the valiie p thermodynamic equilibrium state is sensitive to the choice of boundary conditions and each choice gives, in prindple, a different state in the condensation region although accidental degeneracy is possible. The phenomenon of condensation corresponds to a phase transition of the System and the equihbrium states have less ergodic properties than the states in the single phase (z < 1) region. Although the states occurring in Theorem 5.2.32(B) are invariant under the action T of the group [R^' of space translations, they are no longer [R^'-ergodic. This is most easily seen by calculating that used in the limit. Thus the
lim \X\
>
CD(W(f)r,(W(g}})
=
o,(W(f])cD(W(g]]
(X)
exp{
X
-
c,p,(ß]2 Re
j d''xf(x) j d^^xg(x]^
Thus
co is strongly mixing and [R ''-ergodic, if, and only if, pQ(ß) decomposition of co into [R^'-ergodic states can be explicitly computed finds the following result-
=
,
0. The
and
one
/'27r
/oo where
K(P,PQ)
0),,
=
^
po(^)
a(r(/))
exp
dpK(p,p,]
{-p/pQ(ß}}
/ Jo
Jaco^,(^)
,
and the states cop^r,
are
determined
exp{/cy V^/' (^" f ^V(^) e-^^ j d''xf(x)) } exp{ (/, (l e-ß^)(^ e-P^r'f)/^}+
-
X
+
-
-
In particular the i^''-ergodic states cop, are no longer gauge-invariant. point functions of the states cop^ are given by
Wp,a(/)fl(0))
=C,p
l d^xf(x} fd^'x'^
J
+
and the local densities
J
(2nr
decomposition
is
(271)-^
J d^P^f(p)e-I^P\l
^-^^V
-
by
Po,,.,(ß} -This
readily verified
=
^-^P +
Pc(ß)
with the aid of the two formulas
/"ja^''(^^^+^^^")3.Jo((^'
+
Jo
roo
/
Jo
dx
The two-
e--Vo((2cx)-'/-) )-l/2)
where JQ is the usiial Bessel function, a,b
^
R and
c
>
=
0.
,-c/2 e-
^')'/')
,
Continuous
Quantum Systems.
Thus the variable p occurring in the integral decomposition density of the condensate and the probability measure dp K(p^
I
measures
p]
73
the
determines
possible condensate densities. It remains to describe the thermodynamic limit of the dynamics in the condensation region. This can be discussed from the same point of view adopted in Proposition 5.2.29 but various changes occur. Most noticeably the equilibrium states do not extend to states over the CCR algebra on Z.-([R^). If (cü5 ^cü5 ^(o) is the cyclic representation corresponding to the equilibrium state co obtained by the fixed-density thermodynamic limit with p > Pc(ß), then TtcoC^)'' contains a representation of the CCR algebra over the subspace 6^(ff} CL^(U^). This follows because any element in .^(IR") can be approximated by elements in U^L^(A) in the norm
the distribution of the various
11/11^ and the
/,öf unitary
j d'p \f(p]\\\
lfm' +
unitary Weyl operators satisfy
\\(W^(f] for
=
G
U^L^(A). (Recall
that
>
v
e-^'T'
estimates of the form
W^(g]]W^(h]^^\\
~
-
<
c,||/- ^11^(11/11
3 when
p^(ß]
<
+
+cx).)
ll^ll) Note that the
exp {üH} generated by the Laplacian on L^(W} maps because on ^(R"} is equal to its Fourier transform. Hence U ^(R"), ^(W) a implements one-parameter group of Bogoliubov transfontiations on the CCR algebra over ^(ff). Since any element in U^L^(A) can be approximated by elements ^(ff) in the norm || ||^, it follows that this group extends by strong continuity to a one-parameter group T of *-automorphisms ofna^C^)" äs in Proposition 5.2.29. If T^ is the one-parameter group of Bogoliubov transformations of the CCR algebra over L^(A) implemented by the Dirichlet La placian //A, one can then estabhsh that group Ut
lim
=
coA,(^TfH5)C)
-
(Q,,, n^(A)r,(7i^(B))n,,(C}^^}
AL->OO over ^([R^), where A^ tends to infinity in the explain the nature of this argument it is preferable to first examine the analogous problem in the setting of Proposition 5.2.31 for v > 3. Choosing A, B, and C äs Weyl operators and using the CCRs one readily reduces the problem to the proof that
for
Weyl operators A, B,
sense
and C
of Theorem 5.2.32. To
(/, X^e""^g)
lim A^
=
(X^/^f, X^l^e^'"g)
,
oo
where
^^
=
(l+g-^^A)(1l-e-^^A)-^
and X
=
(i+e-^^)(i-^-^^)-^
.
forms
the
positive operators XA and the spectively, then the proof of Proposition 5.2.31
is based upon the fact that the
But if XA and
bounded
x are
the
positive quadratic
canonically associated with positive selfadjoint operator X,
re-
74
States in
net XA is
exactly
Quantum Statistical Mechanics
monotonically increasing and But in the proof of Lemma lim A*
for all
its
x.
\lt
&L^(n'').
||(;rA
limit, in
5.2.13
we
the
i)^/'(x + i)-^/V-^ll
+
of Lemma
5.2.13, is
=
o
oo
D(X^I-]
Hence, if \l/
e
lim
i)'/V-(^ + i)'^Vll
A>
sense
established that
||(;rA
+
one
has =
o
.
oo
Therefore, the desired result follows from the estimate
\(f,X^e^'"^g)-(X^I^-f,X^l^e''"g)\ < |((;rA + i)'/V,e''"''^((^A + 1)'^' (^ + i)'/')3)l + \((X^ + 1)'/V, (e''"^ e"")(X + 1)'/-g)| + |(((^A + l)'/2_(x + 1)'/2)/,e''^(^ + 1)'/2^)| -
-
+
\(f^(e""-e""^)g]\
.
This result
retracing one uses
can now be used to establish the analogous fixed-density result by the estimates used to prove Theorems 5.2.30 and 5.2.32. In particular the last two estimates given in the proof of Theorem 5.2.32. We omit
the details.
Finally,
emphasize that the quantitative properties of Bose-Einstein quite sensitive to changes of boundary conditions although qualitative features remain unchanged. If one uses dil//dn aij/ boundary we
condensation the
are
conditions with the
activity
<
er
0, then condensation
z^ < l and
occurs
in all dimensions at
a
value of
the condensate is
principally located near the surface of the System. Thus, in the thermodynamic limit at fixed density the equilibrium State, which characterizes the properties of the System near the origin, shows no trace of the condensate.
EXAMPLE 5.2.33. of
d^/dx^
Let
L~(-L/2, L/2)
=
rf^_,^(,)i^]x^L/2 for all l// G D(HL), where er > )^, where given by 2.Q(L) =
selfadjoint
extension
eigenvalue corresponding
[^+.^wl^\x=-L/2 dx
0. The lowest
eigenvalue ^Q(L) unique solution of
tanh(/IX/2)
=
o
of HL is
(In fact,
if L is
decreases
to
-0--
the Gibbs states
normalized
=
eigenfunction IJ/Q
(2//^)'^"{l
+
(^r^
negative
and is
.
has the property that it increases with L to the
lAoW
for
=
l
A is the
;.
The
=o
dx
l
This
and let HL be the
which satisfies
can
asymptotic value a^.
be chosen
äs
sinh(AL)r^^' cosh(;jc)
.
large enough there is a second negative eigenvalue AI (L) which eigenfunction can be chosen to be anti-symmetric.) Thus, are only defined for /.i < /lo(I) < -0-^, and condensation takes place
and whose
Continuous
z
The critical
density p(ß, z^)
p(ß, z,) and
L|I/^Q(O)|^
äs
density
-^
=
is
=
Qxp{-ß(j-}
now
(2ii)-^
<
l
for p <
p(ß, z}
given by
-^
oo
the
thermodynamic
limit state
-
(/, (1 +ze-^"}/(^
-
ze-^")f)/4]
and
o^(W(f)) >
exp{
p(ß, Za).
=
exp{
-
75
j dpz,e-ßP-(\ -z,e-ß^"r'
0 in the limit L
=
I
.
p satisfies
a,(W(f))
for p
z^
--
Quantum Systems.
(/, (1 +ze-l>")/(^
Similar conclusions
are
valid for
-
v
,^e-l'")f)/4}
>
1.
co
at
fixed
5.3. KMS-States
5.3.1. TheKMS Condition In the
previous section we analyzed two C*-algebras which describe the kineparticle Systems and also analyzed the simplest examples of equilibrium states. Now we discontinue this specific analysis and describe instead various general characterizations of equilibrium phenomena. Principally, we investigate the Kubo-Martin-Schwinger, or KMS, condition briefly outlined
matics of
in the Introduction and used in the calculation of the Gibbs states of the ideal Fermi and Böse gases. Our description of this condition was, sketchy and this will be corrected in the sequel. Recall that if ^
selfadjoint operator on , jS Gibbs equilibrium state a
D^, and exp
G
,
{ßH}
hitherto, rather
=^^(),
=
H is
is of trace-class, then the
i:T^(e-ß^A)
,
"(^)^IV^^ satisfies the condition
formally
co(^T,(5))|,^^.^, with respect to the
automorphism T,(^)
This is
a
particular example
=
(ß(BA]
group -
e^^"Ae-^^^
.
of the KMS condition and it
basically
involves two
elements:
(1)
The
analyticity of the functions t\-^o}(Ait(ß}) in a strip, the strip ßif ß > 0, and, approximate commutation of each pair A, and B, within the
0 < Im/ <
(2)
The State
0).
We will choose the
a
precise definition
of the KMS condition which
emphasizes
latter, algebraic, property and partially deemphasizes the analyticity
erty. Subsequently it will be necessary
Throughout
most of
System (^, R, T) based
by 021, T). Moreover,
to extend the
the rest of this
chapter
properties
we
consider
of a
prop-
analyticity. C*-dynamical
the group U and for brevity we denote such Systems let ^^ denote the set of entire analytic elements for T,
on
we
Definition 2.5.20. Recall that ^(^ is a norm dense *-subalgebra of ^, Propo sition 2.5.22, and it is clearly r-invariant, i.e., if y4 G ^T, then it(^) ^ "^T for all
KMS-States
77
[R. We also consider
a l^*-dynamical System (9}1, r), where 9J1 is a von algebra and T a cr-weakly continuous one-parameter group of *-automorphisms of 9JI. The set 931^ of entire analytic elements for T is no longer norm dense in 9Ji but nevertheless it is dense in the ö--weak topology and T-
t G
Neumann
invariant. We
now
analytic
adopt
a
definition of KMS states
elements which has the
Definition 5.3.1. Let
defined to be
(^, T) be
advantage
a
in terms of dense sets of
C*-dynamical System. The state co ^ G [R, or a (T, jS)-KMS state,
r-KMS state at value
a
cD(Aiiß(B]] for all yl, 5 in
phrased
that it is often easy to corroborate.
=
over
^ is
if
(D(BA]
dense, r-invariant *-subalgebra of ^T. ]^*-dynamical System, a state co over 9}I is defined to be a (T. ß)KMS state, for j8 G [R, if co is normal and the above identity is valid for all A and ^ in a cr-weakly dense T-invariant *-subalgebra of 50It. A T-KMS state at value jS -l is called a i-KMS state. If
(9Jl, T)
is
a norm
a
=
This definition has several immediate consequences. 0 is distinct from the other values. In this j5
The value
=
5.3.1 States that
co
is
a
case
Definition
trace-state,
o}(AB]
=
w(BA)
for all ^, 5 G 91.
Conversely, a trace-state is a (T, 0)-KMS state. Ifß^Q and oj (T, jS)-KMS state the definition indicates that T measures the deviation of oj from being a trace. This idea will be clarified later.
is
a
Another immediate consequence is that if t \-^ i:^ then co is a (T, ^)-KMS state if, and
=
i
is the trivial group of
automorphisms,
only if, it is a trace-state. Next note that co is a (T/, j5)-KMS state if, and only if, it is a (T_^/, 1)KMS state. If ^ 7^ 0, this Statement is evident but if ^ 0 both sets of states are the trace-states. Thus by rescaling the group one can eliminate ß. Hence, for many general purposes it sufiices to consider i-KMS states. This also shows that the choice ^ l in the definition of these latter states has no particular significance. (The value -l coincides with a convention adopted in the modular theory of Tomita-Takesaki). We emphasize, however, that despite these re scaling properties there is no simple prescription for connecting the (T, ^)-KMS states for different ß (see Theorem 5.3.35). =
=
EXAMPLE 5.3.2.
Let
^(l))
be the CAR
algebra
over
the Hubert space
I)
and
i
a
one-parameter group of Bogoliubov transformations such that T? ((/)) a(e''^ f), etc. Let/be an analytic element for //, then the i-KMS condition states that =
w(a^(f)a{g))
Therefore
w(a(g)a'(e'l>"f)) ^(g,e-l^"f)-co(a'(e-l>"f)a(g)] =
.
78
States in
Quantum Statistical Mechanics
ay(a'(f)a(g]]
(g,e-l'"(^+e-l^"r'f)
=
.
Iteration of this calculation in the
manner preceding Proposition 5.2.23 shows that gauge-invariant quasi-free state with the preceding two-point function is the unique (T,jS)-KMS state. Note that the polynomials in a(f] and fl*(öf), with/and g analytic for H, are norm dense in ^. In particular the limit Gibbs state constructed in Theorem 5.2.24 is the unique (T,^)-KMS state for the limit dynamics. Ahernatively, let ^ be the CCR algebra over a pre-Hilbert space t), // a positive selfadjoint operator defined on I) not having zero äs its eigenvalue, and such that e'^^!) C I) for all / e R, and l) C D(e-ß"(^ e'^^)"^) for ^ > 0. Let co be the gauge invariant quasi-free state with the two-point function
the
-
co(/)a,(ö)) and
T
the
cr-weakly-continuous
e-^"(\-e-'^")-^f]
(g,
=
group of
of
*-automorphisms
Tia^(^)"
obtained
by
strong continuity from the Bogoliubov transformations determined by e'^^. It follows from the calculation over
I)
the
von
Neumann
preceding Proposition 5.2.28 that co is a (T,j5)-KMS state algebra 7lfo(^)'^ But if ^ is any positive sesquilinear form on
t) with the invariance property
X
s(gj]=s(g,e""f) for all
gjfl) and all point function
t
R, then the gauge-invariant quasi-free
e
co(/K(
(T, j5)-KMS (T,JÖ)-KMS state in a
tensively studied Laplacian and
state
this
case.
n^^C^)'^ An
in Section 5.2.5.
s(gj] where p > 0. It follows
(g,e-P"(i-e-l>"r'f)+s(g,f)
=
over
immediately
=
state with two-
p
.
Thus,
=
j
one
has
possibility
a
of
a
nonunique
is the Rose-Einstein condensation
example Here I)
d\xg(x)
6^(U'')
J
with
d''xf(x)
v
>
3,
H
=
ex-
-V~ is the
,
from the commutation relations
W(f)W(g}=e-^^^^f^^^^W(g}W(f) that the CCR
algebra
over
f) has
a
-(^(/))
unique
trace-state co,
l, =
10,
given by
if/-0 if/^0
This state is invariant under all
Bogoliubov transformations, and thus the group of automorphisms determined by e''^ extends to a one-parameter group T of *-automorphisms of TicoC^l)" However, this group is not cr-weakly-continuous and hence co .
is not
a
(T,0)-KMS
When
ß <0,
state
on
there exists
n^oC^l)" no
in the
(T,^)-KMS
sense
of Definition 5.3.1.
states
over
the CCRs in the
von
Neu
mann sense.
We
begin
the
characteristic of
analysis of KMS states by proving that they satisfy equilibrium, i-invariance.
the crudest
KMS-States
Propositions 5.3.3. Let co be a (T, ß)-KMS the W-algebra 9Ji) mth ß G R\{0}. It follovvs that CD is T- invariant, i.e.,
(D(l,(A]]
the
s täte over
co
is
(T, ß]-KMS
a
state
cD(i:_ißi2(A)iißi2(B]] variant
PROOF.
=
mth
ß
^
(or
(D(A)
=
for allAe^ (for all ^ e äR) and all t 6 IR. Moreover the following conditions are equivalent for over ^ (which is assumed normal in the W*-case}: (1) (2)
C*-algebra
79
an
arbitrary
state co
R.
e
0}(BA), for
all A, B in
a
norm-dense i-in-
*-subalgebra of^-^.
The
proof of the two cases C*, and ^*, is identical. We consider the former. -l. Next let B be an element out by rescaling we may assume ß of the norm-dense i-invariant *-subalgebra S^ C ^^ for which the KMS condition holds and define the analytic function F by First note that
=
F(z) Then F is
an
entire
=
CD(,,(B))
analytic function which X)-
{z;ze C,-l
.
is bounded Imz <
<
on
the
strip
0}
by
M^sup{||T,,(^)||;ye[-l,0]}. This follows because
|F(Z)| But if ^ has
||T,(ß)||
<
identity
an
F(z
i)
-
=
||TRe.(Tn.(5))||
H it follows =
directly from
a;(lT_,.(t.(5)))
=
=
||Tnn,.(B)||
.
the r-KMS condition that
ü;(T,(5)l)
=
F(z)
.
Hence F is
con-
clusion is
now
implies
periodic with period z. If ^ does not have an identity, the same easily reached by use of an approximate identity. But the periodicity
that
|F(z)| for all
z
G C. Hence F is constant
follows than
co
If condition
invariant if
ß
e
ß
=
Liouville's theorem. As S^ is dense in ^ it
is T-invariant.
(2)
of
Proposition
[R\{0}.
0}(ATiß(B)) If
by
< M
=
5.3.3 is
valid,
a
similar argument shows that
co
is
T-
But then
0}(i:iß/2('C-iß/2(A)Tiß/2(B)})
=
Q, this relation is trivially fulfilled. Hence l
Co(T_,-^/2(^)T//^/2(5)) <^
.
2.
It should be emphasized that the value ^ 0, which is excluded in Proposition 5.3.3, is exceptional. If, for example, 91 is abelian, then all states are (T, 0)-KMS =
States but if
T
is not trivial there
One immediate
corollary
System extends in
dynamical PF*-system.
ciated
are
certainly
states which
are
not i-invariant.
of i-invariance is that each KMS state of a
canonical fashion to
a
KMS state of
a
C*-
an asso-
80
States in
Corollary
(^(, T) closure
Quantum Statistical Mechanics
5.3.4.
ivith
ß
SJI^o
Let
co
IR\{0}
G
be
'^(o(^)" of^
=
foUoivs ^''-automorphisms of^R^^
a
all A
Ul and
t G
R. Moreover,
Proposition 5.3.3 implies that unitary representation U,, of IR on ,,j,
7l,(l,(A)) [R
? G
the
C'-dytiamical System of co to the vveak
such that
PROOF.
for all /l G ^l and
of
cyclic representation (^^, TT^^, Q^^). iiniqiie a -\veakly-contmuoiis groiip ?i-^f/ of
it(Tlco(A]] for
state
be the normal extension
co
in the
that there exists
It
(i,ß)-KMS
a
and let
=
exists for each t G U and B G
=
co
(f, ß)-KMS
is
on
93l,,.
is i-invariant and hence there exists
co
a
such that
U,,(t]TL,,(A]U,,(t]-' 2.3.17.
by Corollary
it(B]
Tlco('^t(^])
=
Therefore,
UMBUM''
9Hoj- As Uo, is defined by
U,,(t)n,,(A)ü,,
=
7i,,(i,(A))Q,,
and t^^^t is
strongly continuous, it follows that t \-^U(o(t) is strongly continuous on on ,,j. Therefore, t\-^it is (J-weakly-continuous on 5[R,o- Biit norm-dense i-invariant *-subalgebra of 7if,j(^l) is a cr-weakly-dense f-invariant any *-subalgebra of 9[)lj. Thus, the f-KMS condition follows directly from the i-KMS and hence
7r,,j('5l()Q,^,
condition.
Although
for purposes of verification it is very analytic elements, it is
States with dense subsets of
iinnatural. Thus more
our
next purpose is to extend the
natural class of elements. For this
modulus
principle
Proposition by
5.3.5
(Phragmen-Lindelöf). X)
and T) the closure
o/ D.
=
follows
that the y G
{z ;
on
G
z
Let f be
and bounded and continuous It
one
which is often referred to
a
C,
practical to define i-KMS mathematically somewhat KMS condition to a larger
needs
a
version of the maximum
the three-line theorem.
äs
Let T) be the open
fl
Im
<
z
strip
in C
defined
^}
<
complex function
which is
analytic
on
T),
T).
function
[a, b]
^
g(y}
=
log( sup
\f(x
+
iy)
\xeU
is
convex.
In
sup
particular,
\f(z)\
-
-6l)
It should be
sup |/(.Y -f ia)\, max<^L^elR
emphasized that this result principle because
the maximum modulus
sup_,^
is not
a
^^[/(x
-j-
ib]\
straightforward extension of analytic functions
there exist entire
KMS-States
which
bounded
are
the
on
boundary {z;
Im z G
nevertheless, are unbounded on T) itself. (For theorem, see Notes and Remarks).
{a^b}}
of the
8l
strip 'S but,
reference to the three-line
a
We also need
a version of the so-called edge of the wedge theorem. This originally motivated by problems of quantum field theory and it has many important applications in this latter domain (see Notes and Re marks). The general theorem concerns functions of several complex variables, but the following proposition is only stated for one variable, and in this form it is an immediate consequence of the Schwarz reflection principle.
theorem
was
5.3.6.
Proposition -/^
=
^0
^n[R
Let
and
be
{P C C
an
connected
open
set
such
that
define X)
{z;
=
z
=
+
;c
/;;
6
C,
j >
0}
n
Let F be
a complex function which is holomorphic on 0 for x G i^. Suppose furthermore that F(x) Itfollo\vs that F(z) O/or all z G T).
T)U''/^.
^
.
T) and continuous
on
=
=
After these function theoretic
preliminaries
we are now
ready
to
derive the
first set of alternative characterizations of KMS states.
Proposition System,
ß
eU.
5.3.7.
(^, T)
Let
co a s täte over
be
C"" -dynamical System, or a W-dynamical to be normal in the W*-case, and
a
^ \vhich is assumed
Define DO '
=
{z;
G
z
C,
0 < Im
z
<
j6}
ifß>0and
D^ ifß<0,
and let
T)^
{z;
=
60
is
a
are
i-KMS state
For any
analytic
pair A^ on
T>ß,
B
at
of^ßif ß^(^
value
all
t G
For any
analytic
a
=
Q.
=
co(AT,(B)) =
on
X)^,
B
u'/?zc/z is
such that
,
co(i,(B}A}
[R.
pair A^ on
t G
f)ß=^Uifß
complex function FA.
T)ß,
B
^, there exists
and continuous
on
a
complex function FA. B u'/z/c/?
'f)ß,
such that
,
FA,B(t^iß}^oj(i,(B}A) all
and
and bounded and continuous
FA^B(t)=co(Ai,(B))
for
0}
ß.
^, there exists
FA,ß(t^iß) for
<
z
equivalent:
FA^B(t}
(3)
< Im
C, ^
G
be the closure
The following conditions
(1) (2)
z
[R.
is
States in
82
Quantum Statistical Mechanics
Furthermore
conditions
(2)
if these conditions are satisfied, (3) satisfies the boiind
then the
fiinction FA^
B
^^^
and
sup|F,,s(z)|< \\A\\\\B\\ ze%
andforA G ^l, 5 G ^T, the fiinction identifies entire analytic function z ^^ (jo(Aiz(B]]. PROOF.
(1)
For
e
A,B
G
z
to
T)
of the
^ (2) Let S^ denote the *-subalgebra of ^^ occuring in Definition 5.3.1. ST, define FA^B by
FA^B(Z] for all
mth the restriction
C. Then
FA^
is entire
B
CO(AI.^(B]]
=
analytic,
FA^B(t]
o:>(Ait(B]]
=
for t G R, and
FA^ B(t As
z T-(5) is [0, ^] i-^||T/y(5)|| l>
y G
+
iß)
=
strongly
CD(A^iß(it(B]]]
analytic,
CD(it(B]A)
=
Proposition
.
it
2.5.21,
is continuous, and hence bounded. Define M M
follows
that
by
sup{||T,-,(5)||;ye[0, /;]}.
=
Thus
FA^B(t + for
t +
iy
e
iy)\< IM||||T/X^)|| <M|M||
X), and (2) holds for ^,
5 G
33^-
For
general A,
B e'il choose sequences
{^L>i, {^.}.>i, in S, such that \\A,,\\ < |M||, \\B,,\\ < ||5||, 7i,o(A)ü,,, 7r,,(^;;)Q,, 7i,o(.4*)^co, and 71,0(5,0^0 ^co(B}^a^, 7i(a(B*)ü.^. (In the C*-case this approximation can be made in the ^
^l is
->
^*-algebra, the approximation Now define F by a
is
possible
and and norm
because of Theorem
n,(A,)ü,, 7i,o(B;)Q,, topology. 2.4.16.)
^
-^
If
F(z)=F^,,,B,.(z) for
z
G
T). The three-line
assumes
theorem, Proposition 5.3.3, implies that |F(z) F,(z)| on the boundary of T) and for z on this boimdary one
its maximum value
has
\Fn(z)
-
F,n(z)\
<
max| sup
co(^,T,(5))
-
cü(A,rt(B,n})\
,
UeK sup
\co(T:t(Bn)A)
-
a}(if(B,n)A,n)\
?6(R
<
||5||{||7r,(^* -^;)Q|| +
|M||{||7i(5;; -s:,)a,||
The last estimate is valid for the
T-invariance of
equality.
co
Hence F
sum
established in
is
a
+
Cauchy
\\H,(A +
||;i(5
-^,)a||} -5,)Q,i|}
.
of the terms in the maximum and follows from
Proposition 5.3.3 and the Cauchy-Schwarz insequence
therefore continuous and bounded
on
uniformly
on
T). The limit function is
X, and analytic in D. Also
83
KMS-States
FA^B(t]
lim
=
>
lim
FA^B(t + iß)=
0}(An'Ct(Bn))
=
0}(A'Ct(B))
,
oo
FA^B(t
+
iß]
>oo
lim
-
(D(i:t(Bn)An]
=
a}(it(B}A)
.
n^ oo
(2) (3)
This is trivial
(3): (1):
=^ ^
IfA.Be^r. define
G^,^(z) for all
z
C. Then
e
G^,
is
B
entire
an
co(^T,(5))
-
analytic function and
GA^B(t)=^FA^B(t) for r e (R. It follows from the
GA^B(Z)
=
for all
FA, B(Z)
z
edge of the wedge theorem, Proposition 5.3.6, that X), i.e.,
G
FA,B(Z} for all
z
O}(AI.^(B)]
=
f). But then, by condition (3)
e
(D(Ai:iß(B))=FA,B(iß} for all A,B e ^t, i.e.,
is
co
^
The
(2)
(3)
and
^
co(BA}
(T, ^)-KMS.
The last Statement of the
(1)
=
proposition
was
established
during
the
of
proof
(1).
line of
reasoning allows the explicit extension of a (T, ß)-KMS C*-dynamical System to a (f, /?)-KMS state over the assodated von Neumann algebra 931^0 '^ü}(^T äs described in Corollary 5.3.4. We next that KM condition the S implies that QCO is separating for ^Jloj, but for prove later purposes we isolate the following lemma. State
same
(D over a
=
Lemma 5.3.8. vector, and
all A G
for
PROOF.
Let 9JI be
a C*-algebra on a Hubert space , Q 0 cyclic 0 implies that (D(AÄ''} corresponding state, Ifo}(A*A} ^, then it follows that Q is separating for 9J1.
co
the
Assume that AQ.
(BA^Ü.
esis ^*^*Q
:=
Corollary
5.3.9.
^
at
value
ß
^
defined
If
Let
j5 7^ 0,
0. Then BAQ.
=
(), AsQ. is
U, and
It follows that PROOF.
=
0 for all B G
=
0
9JI, and hence by hypoth=
Q and hence A
=
0.
(^, T) be a C*-dynamical System, co a i-KMS state on (^, TT^J, ^co) the corresponding cyclic representation.
UcoC^}'^.
QOJ is separating for it follows
by Corollary
5.3.4 that the state cb
on
^)l,^
=
Ua^C^}"
by
cb(A) satisfies the A G
=
it follows that A""
cyclic
unit
^loj be
(f, j5)-KMS an
the function
condition for
element such that
corresponding
FA^^A(t}
=
to
group t^-^i{ of
a
a)(A*A)
^*, A,
äs
a;(^*T,(^))
(Q,,, A^,,)
=
=
in
-
*-automorphisms
0, i.e., such that A^^o
Proposition
=
5.3.7. Then
(^Q,, U^(t)A^^}
=
0
0. Let
of SJIoj- Let
FA\A(Z}
be
84
States in
for
Quantum Statistical Mechanics
U. U follows from the
t e
FA^^A(Z)
=
0 for all
with Im
z
of the
edge
wedge theorem, Proposition 5.3.6, that ß. But then
between 0 and
z
d)(ÄA^)=FA^^A(iß) Hence If
^
=
^.
üco is separating for ^co by Lemma 5.3.8. 0, then co is a trace-state on na)(^)" and Lemma 5.3.8 is again applicable. =
The
separating property of Q^o for na)(^)'' provides the prindpal link be theory of Tomita-Takesaki and the theory of KMS states. Assume that co is a T-KMS state over a von Neumann algebra 93^ for some aweakly-continuous group T of *-automorphisms. Thus by Corollary 5.3.9, co is faithful on 7i(^(SR). But the kernel of TT^O is a (j-weakly-closed two-sided ideal 3 C 9JI and by Proposition 2.4.22 there exists a projection E E 9JJn9Jl' such that 3 äR(1] -E). It follows that co(l E) 0, and co is faithful on ^E because 71^0 restricts to an isomorphism between "^E and UcoC^}, and tween the modular
=
=
-
co(^^)
=
co(^)
=
(Q,7i,,(^)Q^).
The remarkable consequence of the Tomita-Takesaki modular theory is that this condition is also sufficient for co to be a T-KMS state for some group T. This
unexpected result follows basically from the identity of the modular condition, described after Definition 2.5.15, and the KMS condition expressed for the modular group. Theorem 5.3.10 normal state The
(Takesaki).
following
(1)
is
co
E G
conditions
faithful
äs
a
are
a
von
Neumann
and
algebra,
co
a
equivalent:
state
9}lnaR^ such that co(1]
There exists
(2)
Let '^ be
9JZ.
on
on -
71^^(9}^), i,e., there exists a projection 0 and co|sjr,j is faithful. E) =
cr-vveakly-continuous one-parameter tomorphisms of^Jl such that co is a i-KMS state. a
Furthermore, if these conditions
are
fulfilled,
the
group
T
automorphism
of
*-t/-
group
T
leaves E fixe d,
'^t(E]=E for
all
t G
[R, and the restriction ofi
restriction is the modular
to
^E is
automorphism
uniquely determined by co. This of^^E associated with co
group
.
PROOF. We have sequence of
already remarked before the theorem that (2) Corollary 5.3.9. As the definition of T on 501(11 E) has
=>
-
the KMS property,
we
may
90t, when proving (1)
assume
(2).
that E
Thus
=
^, i.e.,
we
may
assume
(1)
no
that
is
a
con
influence w
on
is faithful
passing to the cyclic representation associated given by a separating and cyclic vector. Let A be the modular operator associated with the pair (9JI, O), Definition 2.5.10, and let A'^^A"'^ be the corresponding modular automorphism group, Definition (Tt(-^) on
with co,
we
may
assume
=^
that
co
=
2.5.15. Then, for
A,
B ^
m,,
is
KMS-States
a}(AB)
=
=
-
Hence
is
co
a
85
(A^'Ü.'BQ.)
(A^%*Q, A^/^y^Q) (jo(Ba_i(A)) (Q, BM^-^Q) =
T-KMS state in the
.
of Definition 5.3.1.
sense
is a consequence of proof of (1) => (2), and the property T^(^) the fact that co is i-invariant, i.e., T lifts to the representation nco äs described in Corollary 5.3.4. To show uniqueness of the restriction of i to SDZE", we may again e''^ assume that co is faithful and given by a separating and cyclic vector Q. If U(t} is the unitary group on corresponding to T, and ^, 5 G 931^, we have This ends the
:=
=
0}(AB)
=
o}(ri(B}A}
and hence
(A^/-5*Q, A^/-y4Q)
-
(Q, ABQ)
=
(Q, Be^A^)
As 9}!^ is relation
strong*-dense in 50^, it follows impHes that A'/^^Q e D(A'/^*) A/^Q
-
(5*Q, e"A^)
that ^Jl^Q is =
=
>(A^/-)
a core
for
.
A^/^,
and the above
and
e"AQ.
for all ^ E 501^. Now let 901o be the *-subalgebra of '^r consisting of elements with compact Spectrum relative to T. Then ^Q is dense in 931 by Lemma 3.2.39 and since
9Jlo it follows that e^9Jlo^ ^ 9}lo^. But äs any element in ^QÜ is conspectral projection of H corresponding to a compact spectral interval, it follows that ^QÜ. consists of entire analytic elements for e^ Hence ^Jlo^ is a core for e^ by application of Corollary 3.1.20 with S e^ Thus A is an extension of e^. As A and e^ are selfadjoint it follows that A e^, and hence T is the modular automorphism group.
T_/(9Jlo)
=
tained in
a
.
=
.
,
Combining 5.3.3, 5.3.9, and 5.3.10 one sees that a given state co on a C*algebra ^ is a KMS state for a one-parameter group T of *-automorphisms of ^l^a(^]" if, and only if, the normal extension co of co to nüj(^]" is faithful, and in this case the group T is unique. The converse problem of constructing KMS States when T is given is much more complicated and only has a positive solution for special classes of ^ and T. We will partially analyze this problem at the end of this subsection and in the following one. The principal aim in the rest of this subsection is the derivation of alter native formulations and characterizations of the KMS condition. The analysis is essentially identical for C*- or ]^*-systems, and, for simplicity, we present all results in the C*-case and omit the analogous /^*-statements. The first reformulation expresses the analytic properties in a way which is often useful for applications to the thermodynamic limit. For this we need some properties of analytic functions. Recall that if ^ ^((R) denotes the set of infinitely differentiable functions with compact support and if / G ^ the inverse Fourier transform =
States in
Quantum Statistical Mechanics
/oo dp e^^-^f(p] -OO
is
entire
an
5^
function.
analytic
/
G
is
imiformly bounded for
Moreover, in restriction
to
[R
one
has
e^([R),i.e.,/isinfinitelyoftendifferentiable,and;cG[RH^;c^^^'VW/^'"^^
=
all /?,
m
f\l. More
G
predsely
has the
one
following:
Proposition 5.3.11 (Paley-Wiener). A function f is the inverse Fourier transform of a function /G ^ mth support in [R-> R] if, and only if,fis entire analytic andfor each integer n there exists a constant C such that
|/(z)|
"
only need the "only if Statement of this proposition. This is easily established (see Notes and Remarks). Now we are prepared for the first reformulation of the KMS condition.
Proposition 5. 3. 12. Let(^^ T) beaC^-dynamicalsystem,andcüastateover^, The following conditions are equivalent:
(1) (2)
co
the
satisfies
(T, ß)-KMS
condition,
The relation /"OO
/oo
dt f(t}m(Ai,(B])
PROOF.
(1)
validfor
^
all A, B e"^ and all f mth
If j5 G
(2):
Ul^, then
z ^
CD(Ai,(B]) for all
t G
dtf(t
iß)(j4it(B)A)
+
J oo
-OO
is
/
=
/G
co(^T.(5))
^.
is entire
analytic
and
a4i,_,ß(B]A)
=
[R. The function
z^/(z)a;(T,(^M) is entire <
and decreases faster than
analytic ß, by Proposition
/'OO -
5.3.11.
|Re zp"
äs
Re
z
^
oo,
provided |Im
z
Hence, by Cauchy's theorem fOO
dt
f(t)a)(Ar,(B))
/
=
J
OO
f
=
dt
f(t)a)(i,^,ß(B)A)
dt
f(t
OC
+
iß)co(i,(ß)A)
.
./-OC
The condition for
(2)
=>
If
(1):
general B (2) holds
G
'^l follows
and B G
by continuity
and the
decay properties of/.
^I^ then inversion of the foregoing argument
establishes that
/oo
dt
f(t)o)(r,^iß(B)A)
.
-00
Choose
\x\
>
f,, + 1.
e
^
such
that
0 <
/,
<
I,/,(.T)
=
l
if
|.T|
Thus, for any bounded, continuous function
g,
and
f^,(x)
=
0
if
KMS-States
/oo dxfn(x)g(x)
=
87
g(^]
-00
and hence
0}(AB] Replacement
of ^
by iiß(B]
then
cD(i,iß(B]A]
=
.
the i-KMS condition.
gives
Utility of Proposition 5.3.12 can be illustrated by consideration of a sequence T" of strongly continuous one-parameter groups of *-automorphisms of ^ converging strongly to a group T, i.e., ||T"(yl) 0 äs 77 oc for Tr(^)|| The
^
^
all ^ G ^ and
converges in the -^
o}(Ait(B))
and
U. Assume that co
t G
weak*-topology
to a
and hence if /G ^ the
Proposition
5.3.12
sequence of (T", jÖ)-KMS states vvhich state co. It then follows that co(^T"(5))
is
a
Lebesgue-dominated
convergence theorem
that
imply
/oo
/oo
^?/(Ocü(^T,(5))
=
/ 7-00
lim "->
-00
dtf(t)o}(Ai^(B))
/oo J//(?
+
/)ß)a;(T;'(5)^)
-00
/oo dtf(t -
i.e.,
(D
is
weaker
a
(T,jS)-KMS
EXAMPLE
iß)co(r,(B)A},
type of reasoning
same
can
be
applied
under
to the convergence of Green functions.
assumptions
groups of
state. This
+
OO
5.3.13. Let i" be
of
*-automorphisms
a a
strongly continuous one-parameter
sequence of
C*-algebra
^ and co
sequence of T" -invariant
a
States. Assume that the limits
G(A,B-t)=
lim
o}n(Ai';(B))
n* oo
exist, for all A,B e'^ and t G [R, and in particular the weak*-limit co of co,, exists. It follows automatically that the functions G\t^^G(A,B\ t) are measurable but they are not necessarily continuous. If one assumes, however, that the G are continuous and the (Dn are (T'', jß)-KMS states for some ^ G (R, then Proposition 5.3.12 implies that Qfü is separating for ii(^(^]" For this one first applies the Lebesgue-dominated ,
convergence theorem to conclude that rOG
/oo
dtf(t)G(A,B]t)= / J
00
for
all/with /G
^.
\G(A,B-t)\^
Second, lim
<
one
dtf(t-{-iß]G(B,A',-t] 00
estimates that
CDn(AA^)cDn(B^B}=
\\n^(A^)Q,,f\\7i^(B)^^\\~
/;> oo
and hence
G(A,B',t) where
/
1-^^^; is
a
weakly
=
(7r,,(^*)Q,, X,7L^(B)Q^)
continuous one-parameter
\\Xt\\ < 1. Next, for ^ G na}(^)" one chooses^,, ^*^w. Finally one has 7rco(^*)Qü; -^
G
family
,
of bounded operators with
51 such that
no,(An}O.ca
-^
^^w and
88
States in
Quantum Statistical Mechanics roo
/oo dtf(t)(A^Ü,,,X,A*ü,,)=
/ J_^
lim >^^00
.00
/
lim
=
dtf(t}G(An,A:-t)
dtf(t^iß)G(AlAn;-t)
J-oo
"^^
/>00
/
-
J
Thus ifAQ,a
=
r äs
~
iß)(AÜ,,,X-,AQ,,)
.
dtf(t)(A*Ü^,X,A'^)
=
0
OO
t\-^X{ is continuous
71,^^1)'' by
+
0, then
J
and
dtf(t oo
one
must have
A*Q(^
0. But then
=
Q^^ is separating for
Lemma 5.3.8.
The next reformulation of the KMS condition theoretic structure which is
an
emphasizes
the
measure-
inherent consequence of the i-invariance of
KMS States. If
(^, T)
is
a
C*-dynamical System,
the functionals /.i^ and VA defined
and ^
on
co
is
a
T-invariant state
on
"iH, consider
by
/oo dtf(t)(o(A*T,(A}) /OO f(t)o4T,(A)A*)
,
-00
dt
-
.
OO
If
/oo e-''"dE(p) OO
is the
spectral decomposition of the canonical unitary representation (^,.,,71(^,0^0), one has
the
/*oo
/oo -
OO
dt
roo
J
00
Hence JLL^ is to a
^'^
oo
Analogoiisly,
now
(7i,o(A)ü,,, dE(p)n,,(A)ü,,)
Let
(^K, T)
^21, and /f^ and VA th^
by
con-
.
(7l(^*)Q, ^(-/')7l(^*)a)
=
extends
given by
characterize the KMS condition
Proposition 5.3.14. over
=
VA is the measure
dVA(p) We
.
actiially a positive Radon measiire on R, i.e., j^i^ positive functional on Co(IR), which is given by
d^iA(p}
in
1
(7i,,(A}ü,,,dE(q)n^(Ä)Q,,)f(q}
tiniiity
T
(Ti^(A)^,,,dE(q)n,.,(A]ü,)-e'P'f(p)e""'
dp J
implementing
group
be
positive
a
by
means
of these
C^-dynamical System,
Baire
.
measiires on
measiires.
co a
i-invariant state
[R associated with
co.
KMS-States
The follomng conditions
(1) (2)
a I- KMS
is
co
The
are
eqiiivalent:
state at
value
^^ and VA
measures
89
are
ß. equivalent,
with
Radon-Nikodym
deri
vative
'-^(P} dVA for
PROOF.
(1)
=
e-^^
all A ^ ^.
^
(2): Using Proposition
5.3.12
(2)
deduce that
we
/oo
ßÄ(f]=
/
dtf(t}(D(A^^,(A}]
J 00 r^ =
/
dtf(t-i-iß)(D(T,(A}A^)
=
VA(kf)
;
J 00
where
k(p)
=
e~^P, The proof of (2)
Note that condition
(2)
^
(1)
of the last
follows
by
proposition
inversion of the argument.
can
be stated
äs
a
set of
inequalities
(2')
di^^p]
>
e-I^Päti^-p)
because iteration of this latter condition
diiA(p]
>
gives
e-ßPd^A^-p]
>
dliA(p}
and then the
identity dvA(p) diJiA^(p) implies that (2) and (2') are equiva following characterizations of the KMS condition are all expressed in terms of inequalities which stem from a stability property of the KMS states. It is remarkable that these inequalities can be expressed in terms of the in finitesimal generator d of the group T and do not directly involve the global lent. The
behavior of the group. The stability properties which
are inherent in the following characterizations partially explained through the principle of the Gibbs equilibrium states. For simplicity let 51 M, x -matrices, acting on the w-dimensional Hubert space
of the KMS condition maximum entropy for the C*-algebra of all n
^
and hence each state
can
be
=
over
^ is of the form cOp where
a)p(yi)-Tr(p^) and p is state
a
copßf^
density
matrix. In
is defined
particular, by setting
if H
H^ e M and
=
jö
G
IR, the Gibbs
e-ß^
^^^~TT^^(e-ß^} Now
one can
define
an
entropy function iS; S ((D p)
co
G
'
-Egj
--Tr,,(plogp)
.
i>
S(co)
G
[0, log 77] by
90
States in
(The function
Quantum Statistical Mechanics
-
x
log
x
is defined
follows that the Gibbs state is the
by continuity to be zero ai x unique state which maximizes
F(cDp)^ß-^S(cOp)-(Df,(H) This follows
basically
from the
=
0}.
It then
the function
.
convexity inequality
-Tr(.4 logA-A log B)
Tr(yi
<
-
B)
for Hermitian matrices A and B and the identification
F(cü,)
=
-r' Tr(p log
p
-
p
log
pßff)
r' log Trg,.(e-^'')
+
(see Chapter 6, Section 2.3). Thus the Gibbs state is characterized by principle of maximum entropy at fixed energy. The maximum entropy principle leads to various stability criteria. For ample, if cp is any map from E^n to E'sii, then
F(co,^J
>
F((p(co,^^))
the
ex-
.
The
simplest illustration of this rule occurs if (cp((D))(A) o}(WAU) for some unitary [/ G ^. For this transformation one has S((p((D)) S(CD), and hence the inequality for F gives =
-Tr^^XP^//^) which
can
be
rephrased
>
-Tr^.(P/?//^*^^)
äs
oj,^,^(U*[H,U]) But the
dynamical
group
T
corresponding
rt(A} and the infinitesimal generator (5 of
ö(A) Thus
one
=
T
=
to H is
given by
e^^^Ae-^^^
by
[iH,A]
.
concludes that the Gibbs state satisfies the
r^co([/*^(t/)) for each
0.
>
>
stability requirements
0
unitary
U e^.ln Section 5.4.4 we will demonstrate that this criterion the fact that Systems in equilibrium are unable to perform mechanical expresses work in cyclic processes. For this last reason states which satisfy the criterion
said to be
passive. In Theorem 5.3.22 we also demonstrate that passivity of large extent equivalent to the KMS condition. The two conditions cannot be completely equivalent because the passivity criterion carries no reference to the temperature, i.e., the value of ß~^ and convex combinations of passive states are passive. A second type of stability criterion follows from consideration of continuous semigroups t >Q ^^Tf such that TfE^^i C E<.ii. If y denotes the infinite are a
state is to
a
,
simal generator of T, then the maximum entropy
principle gives
KMS-States
r'(F(a),^J-F(r,a;,^J)
9l
>0
and hence
-F(ya)p,J For
an
of this latter
example
principle
0
>
.
consider the operator y^; ^
i-^
^ defined
by
ys(A)=B^AB-{B^B,A}/2 for
some
fixed B e^.
Clearly,
jß is bounded and
75(1!)
=
0. But
one
readily
calculates that
yß(A^A) and hence 7^ is
-
yß(A^)A-A^yß(A)
dissipative, by Proposition 3.2.22,
\\A-o^yß(A)\\>\\Al by
[A, 5]*[^, B]
-
Lemma 3.1.15.
and
0,^6^,
>
a
Therefore, t
/
0
r >
r,
H^
=
expj^y^}
=
lim h n
is
> 0
>
oo
-
72
\
\~"
-y^ /
continuous
semigroup of contractions. Moreover, r/H ^=11. It Corollary 3.2.6 that the T/ are positive, i.e., r/iK+ C ^+ and 77^21 ^ E^ for all / > 0. Application of this type of semigroup and convexity arguments leads to a set of inequalities for the Gibbs states which are sometimes referred to äs auto-correlation lo\ver bounds. These inequalities have been proved to be equivalent to the KMS condition. a
uniformly
then follows from
Theorem 5.3.15
(Roepstorff-Araki-Sewell).
The following conditions
(T, ß)-KMS
(1)
co
(2)
-ißo}(A''ö(A))
is
for
a
all A e
>
D(ö),
co a
(1)
=^
(2):
co(^M)log(co(^*^)/co(^*))
W
W-
+ 00,
W
X
>
0,1; > 0, 0,1; > 0, 0,i;= 0
.
U+\-^S(u,v) =u\og(u/v) convex
S(^u, ^v) R+.
>
0,
A routine calculation shows that the function
semicontinuous, jointly
e
C"" -dynamical
51.
state
in
(w, t;)
and
(w, ü), i.e.,
for W, t;, A
a
where
u,veU+ is lower
be
state over
equivalent:
are
{u\og(u/v), PROOF.
(^,T)
Let
System, ö the infinitesimal generator of T, and
Hence it follows that
=
2.S(u, v)
homogeneous
of
degree
l in
States in
Quantum Statistical Mechanics
si^ ^-w/,^ ^iVi \
i
^
<
l
J
i
^iS(ui,Vi)
i
for all finite sequences {/l/}, {w/}, and {i;/} in bounded non-negative functions on [R, and j^i is a
1R+. If f\ and /2 are continuous positive finite Baire measure on [R, it follows from Proposition 4.1.1 that /i can be approximated by measures of finite Support, and the lower semicontinuity of S implies.
5(K/l),K/2)) <^(5(/,,/2)). If
co
is
a
(T,^)-KMS
state,
d^A(p) dvA(p) äs
may form the
we
measures
(n^(A)Q.,,,dE(p)7i^(A)üa^)
=
in the introduction to
Proposition 5.3.14, unless ß inequality in (2) are the above inequality then imply
5.3.14 and
^(co(^*^),co(^^*))
where
we
=
S(ii^(\), v^l))
=
S(^lAl]. ßA(k-']]
<
M^(^(l,^-'))
=
M^(logyc)
-
-iß(n^(A)ü^,iH^n,,(A]^^)
have used the notation
To prove
(2)
(1)
^
Lemma 5.3.16.
generator
ofi,
It follows that
need the
we
(^, T)
Lei
and
be
a
that
assiime
0)
U(^(t)
=
e'^^"^
following
0, in which
=
trace-state, and both sides of the
Proposition
,
(n^(A*)üa,,dE(-p)n,,(A*)^^)
=
Define
zero.
case
k(p)
-iß(D(A'^d(A])
=
is
co =
a
e~^P
.
,
.
lemma.
C*-dynamical System, let d be the infinitesimal CD(Ad(A]] G M for all A ==^* G D(ö).
is i-invariant, i,e.,
CO(T,(^))-CO(^) for allAe^ PROOF.
As
and
ö(AY
t G
=
R. for A
^(A)
a}(ö(A)A) where the last
equality
is
a
A* ^
=
=
it follows that
D(ö)
co(AÖ(A))
=
-co(AÖ(A))
consequence of the
CD(Ö(A~))
=
co(ö(A)A)
+
,
of the lemma. Hence
hypothesis
co(AÖ(A))
=
0
,
and it follows that
CO(T,(^-))
-
oj(A^)
l
=
ds(D(d('L,(A)^))
=
0
.
JQ
The T-invariance then follows
by continuity
2.2.11, i.e., each element of ^ is END
OF PROOF OF
a
THEOREM 5.3.15.
bounds and Lemma 5.3.16
imply
from
Proposition
linear combination of four
(2)
that
co
^
(1).
If
^ 7^
2.2.10 and Theorem
positive
elements.
0 the auto-correlation lower
is r-invariant and
we
may form the
measures
KMS-States
^^ and VA for all A e ^. If
^ it follows
/G
essentially
from
Propositions
93
5.3.11 and
2.5.22 that
if(A] is entire
analytic
for d
=
=
=
=
j dtf(t]T,(A]
and, in particular,
t/<(/)
Defining h(p)
=
\f(p]\^
and
if(A)
G
D(d].
Now
/ dtf(t)U,,(t) y dtf(t} j dE(p)e-"" j f(p}dE(p] f(-H^)
we
compute
=
k(p)
=
e~''''
we
.
deduce further that
-ißwMAYöMA)))
By
similar
=
-iß(U(f)n^(A)fl,,, m^ U(f]Ti,,(A)^^}
=
(n^(A)Q,J(-H)ßHj(-H^)n,,{A)Q)=ß^(\og(k)h).
computations,
one
finds
w(Tf(Arzf(A)) Co(Tf(A)Tf(AY) -ißw(Tf(A)5MAr)) Thus the auto-correlation lowerbounds,
=
n^(h)
,
=
VA(h)
=
-v,(log(^)A)
,
applied to T:/ (A)
/i^(log(yt)//)
>
S(^A(h). VA(h))
-VA(\Og(k)h)
>
S(VA(h]^ ^A(h]]
.
and
T:f(AY, respectively, give
,
Now, define p(h) sup(supp h) and p(h) inf(supp h). Assuming for the that j5 > 0, we deduce from log k(p) -ßp that =
=
moment
=
-ßp(h)h
hlogk
<
The auto-correlation lower bound then
or,
<
-ßp(h)h
implies
.
that
-ßp(h}^A(h)
>
ßp(h)vA(h]
>
VA(h)\Og(vA(h)/fiA(h}).
e-ßP^^\AW
>
^A(h}
^AW^Og(^AW/VAm
,
equivalently >
e-ßP(^^^VA(h]
.
Also, since
e-f^P^^^h
>hk>
e-f^P^^^h ,
we
have
e-^P-^^^^^A(h] Let e>0 and let
^^^ hn
=
l
{/z}^>i
pointwise,
be
>
a
VA(hk)
>
e-ßP^^^VA(h]
sequence of
positive elements
and
\^-ßp(hn}_^-ßp(h}^
.
^ ^
in Q> such that
94
States in
Quantum
Then the above
Statistical Mechanics
that
inequalities imply
\llA(hhn]-^^A(hhnk]\ and so,
<
V^(/2/Z,)
by Lebesgue's theorem,
|A(,(/2)-v^(M)|
<
,VA(h].
Hence
^iA(h) for all h e ^. But this is
Theorem 5.3.14. When
Finally
if
^
ß
just <
=
VA(hk)
the measure-theoretic form of the KMS
0 ihe
is the
reasoning
0 the auto-correlation lower bound
=
for all A e^; thus
by interchanging
A and
condition,
same.
implies
A*, co(A*A)
that
co(^^*),
=
o}(A*A} and
o;
<
(Jo(AA*)
is
a
trace-
state.
We next derive
characterization of the KMS condition
a
"auto-corre
by
lation upper bounds." This is formulated in terms of a quantity which occurs in a perturbation expansion of the free energy, and is known alternatively äs the Duhamel
product, a
two-point function, Bogoliubov or
the canonical correlation. If
T-invariant state
on
the entire
(A,B)^
=
-
P
on
analytic
[
51, this quantity is elements ^^ by
d^o3(A^Ta(B))
-
scalar
product, Kubo-Mari scalar (^, T) C*-dynamical System and CD a priori defined äs a sesquilinear form is
a
(n^(A)Q,,,(ßH,,r' (^
e-ß^-}n^(B)Ü^)
-
JQ
for A,B e ^T. Here and later
we define p\-^p~^(l e^P] by continuity to be 0. equal to one when ;? Formally, this quantity appears in the second-order term of a perturbation expansion of the partition function. Let us reconsider the example ^ M and =
=
the Gibbs state
Tr(e-/'^^)
-M^)-T|:M^,^,
The
quantity
Z(^//)=Trs(e-'''^) is
usually referred
ß\A,B)^
=
ß'~
to
l
äs
dt
the
partition
function and
one
computes that
cDßH(A^^uß(B]]
Jo =
Tr(^^*
/ dte-ß^^ßBe-ß^^-^^^}/Z(ßH) JQ
-
f dt 4-Tr5( dy
l
/z(^//)
e-W^-^^)/J^*e-/'(i-0(//-^5))
JQ
^'.Trs(e-^-^--^^))_^^/Z(^//)
dxdy
^^Z(ß(H-xA*-yB])^^^^Jz(ßH).
'
/
KMS-States
The
canonical correlation
name
derived in
a
similar
from the
comes
following
95
formula which is
manner
^O^ß(H-.B}(Ä}l
=
ß{(A\B)^-COßH(A}cDßHm
,=
Subsequently, in Theorem 5.4.12, strongly clustering, i.e.,
show that if
we
is
co
-
(T,/?)-KMS
a
state
which is
Um
(jo(AT:t(B}}
=
CD(A]o}(B}
/>oo
for
MA,B
^, then
e
[
iß(A,B)^^ißcD(A)a}(B)-i-\im r->00
dtco([A,T,(B}]),
JQ
The last term represents the first-order change of a)(A} when co is allowed to envolve under a perturbed dynamics, with perturbation B. Thus it is often called the linear response. This is fully discussed in Section 5.4. The inequahties occurring in the following theorem will be referred to
äs
the
auto-correlation upper bounds.
Theorem 5.3.17
(Roepstorff-Fannes-Verbeure).
be
(^, T)
Let
a
C'^-dyna-
mical System, and CD a i-invariant state over 9l. The following conditions are equivalent:
(i,ß) KMS state f^ ß-^ / J/ICO(^*TU(^))< (a;(^*^)-co(^yl*))/log(co(yi*yi)/co(yi.4*))
(1)
o;
is
a
-
,
(2)
Jo
all A G
for
^Ir, where
define
we
(u
(u
-
v]/\og(u/v]
We consider
PROOF.
only
<(
=
the
v]/\og(u/v],
W
>
0,
W,
W
=
z;
0,
uv
case
-
j5 7^
0 and leave the
=
z;
>
>
0,
0
0,
W
/
u,
.
marginal
case
ß
=
0
äs an
exercise.
(1)
If ^ e
(2):
=^
Proposition 5.3.14, condition implies
^T define the
and
again
define
measures
k(p)
/.^(r')
=
=
/i^ and VA
äs
in the introduction to
e~f^P. By Proposition 5.3.14 the KMS
vxi),
while the auto-correlation upper bound states that
/i^((l -Ä:-')/logÄ:) Hence
we
a
(v^(l) -/z^(l))/log(v^(l)//i^(l))
.
must prove
/.,((! By
<
change
-r
')/iogfc)
of scale, it is
<
(/.^(r')-^,(i))/iog(/.,(r')//.,(i))
enough
to prove that
.
96
States in
Quantum Statistical Mechanics
[d^l(p)p-\e>'-\)
(
<
J
l d^i(p)e''- fdn(p)]/log(fd^i(p)eP/ j dn(p)\
\ J
for any
Define
'J
positive finite Baire a function/by
measure
f(t)
=
I.L
l
/
on
\J
R such that
log(fd^i(p)e"'} V'-'
This function is
convex
in
/"(O
=
Jdi.i(p)e^P
<
/
for all
oo
.
/
j dn(p]pe"'/ j dn(p]e"'
=
,
(^ l dn(p)e'^ j di.i(p]p^e"' ( j d^i(p)pe"']'^ j (j rf^^/')^"')' -
where the last
inequality
R.
t e
because
t
f'(t]
J
is
a
consequence of the
> 0
,
Cauchy-Schwarz inequality,
In
particular,
/(/)< (l -0/(0)
//(!)
+
and hence
/
rf?exp{/(0}<
JO
/
ö'/exp{/(0)}expW/(l)-/(0))}
JO =
(exp{/(l)}
-
exp{/(0)})/(/(l)
-
/(O))
.
But l
dt ex|: xp{/W}
=
0
and
so we
obtain the desired
inequality
ld^l(p]p-\ef-\]< ( l d^i(p)-
J
\J
(2)
=
(1).
/rfM;')p"'K-i)
l'd^i(p)ep}/\og( fd^t(p)/ fdn{p)eA
J
/
/
\J
l
J
.
/
We will prove that the auto-correlation upper bound implies the appeal to Theorem 5.3.15. Again we may
auto-correlation lower bound and then assume
j5
=
l
by changing
scale. Consider the function
f(p] It follows from the
=
p-\eP-\]= l
integral representation
that
dte^^
/
is
.
strictly increasing
and convex,
and hence the inverse function g,
g(p-\e''-\)) exists
on
IR"^ and is strictly increasing and
Defining
x
=
co(ÄÄ*)/co(A*A),
=
p,
concave.
the auto-correlation upper bound foT
the form
d^^(p)p-\eP-l)/CD(A^A) J
and hence
<
(x-l)/\ogx
ß
=
l takes
KMS-States
cj( But
äs
g is
concave
d^i,(p)p-^(eP-l)/o,(A^A)]
and
continuous, and d iJ,^(p) / w(A''A] is
a
97
.
probability
measure, it
follows that
g(j d^,(p]p-\eP l)/co(^*^)) j d^i^(p]g(p-\eP J^ d}^^(p)p/co(A^A) >
-
-
1))M^M)
=
=
(n,,(A)ü,,
=
ico(A''ö(A})/a}(A''A)
-
H^n,,(A)Q,,)/co(A^A)
and hence
io}(A''ö(A)) both sides
Multiplying As
^IT is
-l
co(A'^A)\og(o}(AA'')/co(A*A))
.
obtains the auto-correlation lower bound ai
one
ß
=
1.
for the infinitesimal generator ö of T, by Corollary 3.1.7, the inof Theorem 5.3.15 follow for general A G D(ö) by continuity.
a
equalities
by
<
core
In Order to
proceed further in the analysis of KM S states, and in particular passivity äs a criterion for the KMS property, we must broaden slightly the latter notion. In statistical mechanics the parameter ß which occurs in the definition of a (T, j8)-KMS state is proportional to the inverse temperature. Thus ß 0 corresponds to infinite temperature and the to examine the condition of
=
associated KMS states, the r-invariant traces, are therefore called chaotic States. At the other extreme one has the zero-temperature case ß + oc and =
the definition of
KMS state extends in
natural fashion to these states.
They corresponding representation, or ground states. One can also formally consider negative temperatures and the extreme case ß -00 and for contrast the corresponding states are called ceiling states. are
a
a
states of minimal energy in the
Definition 5.3.18. and
CD a
Then
Let
(^, T)
be
a
C*-dynamical System,
ö the generator
co
is called
a T
ground
state
if
-icD(A*S(A})
>
0
for all A G
D(ö). In this case CD is also called a i-KMS state at (T, 4-oo)-KMS state. Similarly co is called a T ceiling state if
icD(A^Ö(A)} for all A a
(T,
-
D(ö), and in (X))-KMS state.
There
G
are
ceiling
this
case co
many other ways of
of them in the next for
oft,
state over ^.
states.
value + oc,
or a
> 0
is also called
a
r-KMS state at value
ex, or
characterizing ground states. We collect some proposition. corresponding proposition exists, of course, A
98
States in
Proposition State on
Quantum Statistical Mechanics 5.3.19.
Let
The following conditions
(1) (2)
be
(W, T)
is
(D
and let
co
be
a
equivalent:
are
groiind s täte. ^'^^, then the
a I
If A,
B
is
iiniformly
For any
Imz >
B
analytic fiinction
entire
(D(AI^(B))
zf-^
(3)
C'-dynamical System,
a
^.
bounded in the region {z; z G C, Im z > 0}. ^, there exists a fiinction FA^B ^vhich is continiioiis in
A^ 0, analytic and bounded
in Imz > 0 such that
FA^B(t]=o^(Ait(B]] for all t^R, If f is a fiinction
(4)
C
(-00,0),
mth
Fourier
co(^f(Ar^f(A]] for
(5)
co
all A
G
transform /G ^,
and
supp/
then
=
0
^l.
is i-invariant, and
if
e^^^-7i^(A)^,,^7i,,(i,(A)]^^ is the
corresponding iinitary representation ofR H^> 0
Ifthese conditions
PROOF.
(1)
=^
We will prove (5). If co is a
are
(1) 44> (5), (4) ground state
e'^^-. But 7io,(D(ö))Q^
may form
C
eD(ö).
(5)
ij/ =>
G
n(o(D(ö))Üa}
D(H,,}
(1).
is
=
a core
eD(ö),
TLa^(W)" for
D(H,,)
all
and
=
-i(n,,(A)Ü,o, 7ia,(ö(A))ü,,)
=
-ia}(A'^ö(A})
>
0
7r(T,(D((5)))Q,,
for H,,,
=
by Corollary >
n,,(D(ö})Q,, 3.1.7. It follows that
0
.
This is immediate from the relation
-/co(^*^(^)) foT A
G
=
t e
R
.
(5), and subsequently (5) ^ (3) ^ (2) ^ (5). co is i-invariant by Lemma 5.3.16, and we
(lA, //,,i//) for all
then e'^^'"
But
e'^^-7i,,(D(ö))ü^ and hence
then
then
(n,o(A)Q^, H,,n,,(A)Ü,o)
for alM
^,
.
satisfied,
^
on
(n,.,(A)Ü,, H^n,,(A)ü,,) ,
KMS-States
(5)
If
(4).
==^
is T-invariant
0}
K^(if(A))Q^ by
=
the calciüation used in the
and supp
/
C
(-00, 0)
then
99
have
we
U^(f)n,,(A)Qo.
proof of (2)
(1)
=^
f(-H^)n,,(A)Q^
=
in Theorem 5.3.15. Hence if H^o >Q
0 and thus
f(Ha))
co(v(^)*v(^))
=^
ll^^vW)^-!!'
0
-
for all Ae^.
(4)
(5).
==^
If
(4)
is true it follows from the
o}(A'Cf(B)}
=
Cauchy- Schwarz inequality 0
/G ^ with supp /C (-00, 0). Replacing approximate identity for ^, Proposition 2.2.17, we obtain
for all A, B E 51, and all of
an
co(v(5)) for all B e^^ and all
complex conjugate that 0
^
supp
/.
^ with supp
/G
of the last
co
(5)
(3).
=4>
the elements
IfB
5* it follows
=
by taking
the
that
implies -
(D(B)
is i-invariant. It follows therefore that
(n(A)Q^,f(-H^)n^(B)Q^) whenever supp
by
relation, that the relation still holds for all fe ^ such
But then Lemma 3.2.45
[R, i.e.,
^ G
A
0
/C (-00, 0).
CO(T,(5)) for all
-
that
=
(^*v(5))
=
0
/C ( oo,0). Hence //M > 0 By spectral theory, we may define .
FA.B(Z}
=
(^cu(^')ßa e'^"''n^(B)a^)
=
/
e-'^P(n^(A*)Sl^, dE(p)n^(B)Sl^)
J 00
for Im
>
z
0, where rO
e^^"-
=
/
e-^^PdE(p)
J oo
spectral decomposition of e'^^'". Lebesgue's theorem now implies that FA. B has properties. (3) =^ (2). This is proved by using the edge of the wedge theorem, Proposition 5.3.6, äs in the proof of (3) ^ (1) in Proposition 5.3.7. (2) =^ (5). By an extension of the Phragmen-Lindelöf theorem, Proposition
is the
the desired
5.3.5,
we
have that
a;(^T,(5))| when Im
z
>
0. Hence,
replacing
A
by
the elements of
h(T,(5))| when Im
CD(T:-,(B)) rem,
co
is
z =
>
0. But if B
co(TX5)),
=
IMII ||5||
<
<
and hence
=
approximate identity,
||ß||
B* it follows from the Schwarz reflection
|co(T^(5))|
<
\\B\\
for all
T-invariant, and we can form Hfa. i/^ Tio,(B]^^ is analytic for H^ and
If B e^r then
an
z
G
C.
principle that By Liouville's theo
100
States in
Quantum Statistical
|(,A,e'--""iA)| for Im
z
>
0.
=
Mechanics
|a;(5*T,(5))|
<
Therefore, taking the derivative
co(5*5)
at 0 in the
=
(i^, i/.)
direction Im
z
>
Re
0,
z
=
0,
obtain
we
(lA, -//.o'A) But
Tia)^^-,)^^
is e'^^'" -invariant and thus
0
<
a core
.
for HCO
by Corollary
3.1.7. It follows
that
//o. The last Statement in the
EXAMPLE 5.3.20.
Let
>
is
proposition
UI(I))
0
.
a
be the CAR
conseqiience of
algebra
over a
Corollary
3.2.60.
Hubert space
one-parameter group of Bogoliubov transformations such that
f) and
T
a
T^(fl(/)) a(e'^^f), imique (-i,ß)-KMS state for all
etc. In
=
Example 5.3.2 we have shown that there is a ß ^ U and we next argue that there is a unique T ground state (T ceiling state) if, and only if, there are no nonzero / G l) invariant under the unitary group U^ e''^. First, assume there are no invariant / and let co be a ground state. It follows from Lemma 5.3.16 that co is t-invariant and then it follows äs in Example 5.2.21 that co is even. Now by linearity, anti-linearity, and positivity
co(a*(/)a(3)) where 0
< 7 <
commutes
1
.
The i-invariance
strongly with H.
successive choices A
=
a(f] TH
But the and ^
=
implies that T commutes with Ut and hence T ground-state condition -zco(^*(5(^)) > 0 with the
*(/),/
=
HT
TH
,
These conditions
uniquely determine spectral family of H. In particular
7i,,(a((i
-
(ö,r/),
=
T)f}}ü^
r
=
by
D(H], yield
e
=
T
HT
=
.
'//(-oo, 0)
7ü,(*(r/))n,
0,
the conditions
=
where EH denotes the
0
.
Therefore,
cD(a(g^}a(g2})
=
=
co(a((^
T}gi)a(Tg2)) -co(a(Tg2)a((^-T)g,))=0. -
Moreover,
co(a%fi)a*(f2)a(gi)a(g2)} =
=
=
co(a^(Tf,)a'(Tf2}a(Tg,)a(Tg2)) -cD(a^(Tf,]a(Tg,]d\Tf2]a(Tg2)) -(32. r/2)(^i, r/l)
+
+
(g^Jf2](g2Tf,]
(^1, r/2)(^2, r/l)
of the CARs. A similar calculation for higher-order monomials proves that co unique gauge-invariant quasi-free state with the above two-point function. Conversely, assume that Utf f with / 7^ 0. Let /^ denote the orthogonal complement of / in l). It follows from the construction in the proofof Theorem 5.2.5 that ^(f)) ^ ^(C/) (g)^(/^). Thus, if co/ is any state over ^(C/) and coo is a ground state for T restricted to ^(/^), then co/ 0 COQ is a ground state for T. Thus the ground state is not unique.
by
use
is the
=
101
KMS-States
After this diversion
on
states and
ground
which
ceiling states,
correspond
oo, we return to the discusintuitively to T-KMS states at values -h oo, and sion of stability properties and the KMS condition. So far we have derived
characterizations
various
ß
[Ru{ oo}. ß G [0, oo]
G
for
the
of
some
dition is the
for
condition
(T, jS)-KMS
passivity
a
condition which
condition introduced in the discussion of
preceded general C* -System together notion, complete passivity. Let
Definition 5.3.21.
be
(91, T)
and the
stability
Theorem 5.3.15. We first
maximum entropy principle which precise definition of passivity for a ment of this
fixed
some
implies the KMS property but does not determine the precise value of ß. This con
We next consider
-io)(Wd(U))
co
a
give
be
state
a
a
refine-
where ^ has
C*-dynamical System,
a
identity, let ö be the infinitesimal generator of T, and let Then CD is said to be a passive state if
with
on
an
^.
0
>
for any U G ^o(^)'^^(<5), where ^o(^) denotes the connected component of the identity of the group ^(^) of all unitary elements of ^ with the uniform
topology. Moreover, state of the
The
is said to be
CD
a
completely passive
C*-dynamical System
principal
state
((X)|'^i^, (X)J^^T)
characterizations of
passivity
if
(X)"^ ^
co
for each
is
contained in the
are
a
passive
G f^.
following.
(Pusz-Woronowicz). Let (^, T) be a C^'-dynamical System, identity, ö the infinitesimal generator ofi and CD a state on ^. Let ö^"^ be the generator of @^.^^T: on @"^i^. Consider thefollowing conditions:
Theorem 5.3.22
where ^ has an
(T, ß)-KMS
(1) (2) (3) (4)
CD
is
a
CD
is
CD
is
completely passive. passive.
(5)
-icD(Aä(A]]
state
-i(@]^^cD](Bd^''\B]]
The following
>
for
>
implications
are
j8
0/or
all A
Ofor
a
=
G
all J5
A* G
valid:
and
[0, -f oo],
5* G
-
CD
is i-invariant,
D(ö^''^}
and all
G N.
D(ö).
(1)
(2)
<^
<^
(4)
=^
(3)
=^
(5).
Furthermore, if there exists a group G and an action ÖL of G äs *-automorphisms of^ such that CD is (^.-invariant, a commutes with T, and CD is weakly
a-clustering, i.e., =
a)oc(,g ocgit^rto^g
,
cD
g
,
g eG
eG,t ^R
,
,
and inf \CD(AB') 5'6Co(aG(5))'
then it follows that
(5)
=^
(1)
~
CD(A)CD(B)\
-
0
,
A,B ^^
and all the conditions
are
,
equivalent.
102
States in
Quantum Statistical
Mechanics
Remarks
(1)
Since conditions (3) and (5) are preserved under convex combinations of States, and convex combinations of (T, jß)-KMS states for different ß are not KMS states, it is clear that a condition of purity of CD is needed for the implications (3) =>-(!) and (5) =^ 1.
(2)
If
is
(T, ^)-KMS
state for a ß (0, oc], then CD is automatically by Propositions 5.3.3 and 5.3.19, but this is no longer the case for ß 0. Condition (5) implies invariance, however, and thus the condition of T-invariance cannot be dropped from (1) (see Lemma 5.3.16). co
a
T-invariant
=
PROOF
(2). If co is an invariant KMS state at value ß for (^, T) then it follows that a ^-KMS state for ((X)"^ j^, (X)"^ ,T) and hence (1) =^ (2) will follow from (1) ^ (3). (2) ^ (3). Trivial. (1) =^ (3). When ß G (0, +00) this is an immediate consequence of the autocorrelation lower bounds, Theorem 5.3.15, and when ß +(X) it follows from the definition of a ground state, Definition 5.3.18. The remaining case is 0, i.e., co is a ^ (1)
=^
(X)"^ ,co is
=
=
T-invariant trace-state. Assume first that U e D(ö) is a unitary such that ||t/ H || <2. Then it follows from spectral theory and an extension of Theorem 3.2.32 that there existsan^ A' e D(ö)v^ith\\Ä\\ < TI such that [/ e'^^. But Lemma 3. 2. 3 1 that -
=
implies
=
r\
ö(U)=i
/ dre^'''^ö(A)e^^^-''^^
Jo and hence rl
/ dro}(e^^''~^^^ö(A)e-^^''-^^^)
co(U^Ö(U})=i
ia}(ö(A))
=
=
0 ,
Jo
where the second step relies upon the trace property of from the t-invariance.
Next, if -/co(t/*(5(^)) element such that
>
0 for
||F- U\\
<
some
U G
^f/(^l) r^D(ö)
=
VU'^
G
and the third step follows
and F e
^?/(^) r^D(ö)
is
an
2, then V= UiU
where Ui
co
^?/(^)nD((5)
and
,
\\Ui-^\
\\V
=
-
U\\
<
2. Thus
co(t/i*^(^i))-0 and
-zco(r(5(F))
Hence, the and
thus
=
-i(jo(U*U^ö(UiU))
=
-ico(U"'U;ö(Ui)U)
=
-icD(U;ö(Ui))
set of U e ^/(3I) r^D(ö) such %(5I)n/)((5) is contained
('?/(5I)nD(^))o by
the first part of the
-
that in
-
w4U''UlUid(U]]
ioj(U''^ö(U))
>
0
.
-io}(U"'ö(U))
this
proof )
set.
> 0 is open and closed, (Note that ^/o(^)nZ)(^) =
103
KMS-States
(3)
If y4
(5).
=^
D(ö),
^* G
-
then
e^'^ %(9I)nD(^) for all
e [R
by
Lemma
3.2.31, and r\
d(e'^)
=
i
/ dre^^''^ö(Ay^^^-''^'^ 7o
.
It follows that r\
-/co(e-'^^(e'^^))
/ dro}(e'^^''-^^'^ö(A)e-^''^''-^^A)
-
JQ
r\
-^/Vo 00
co
/
1
^^E
/
yz ^
,, !
,fr^
('_;y'p/J+l
[^...[^,^(^)]...]
U^Vi
2 \
/
-L^KhK-M>(^)i-i) -8co(^(^))+^/co([(5(^),^]) But the
of
passivity
Therefore,
one
must
CD
implies
have
that this
co(d(A]]
=
0(-^)
+
is
expression
0 and
.
for all
nonnegative
i(o}[d(A], A]]
>
0 for all A
=
e R.
A^ e
D(ö).
But then the relation
[Ö(A}, A] implies
=
Ö(A)A
-
AÖ(A)
-
Ö(A')
>
0
-
2AÖ(A)
that
-io}(AÖ(A)) eD(ö).
for all ^ =A*
special case of (3) ^ (5). remaining proofs, namely, (4) =4> (1) and (5) =^ (1) under the weak-clustering assumption on co. We first explore some consequences of (5). IfB e D(ö), then (B + B*)/2 and (B 5*)/2/ are selfadjoint elements in D(ö), and condition (5) implies (2)
^
We
(4).
This is
a
turn to the
now
-
~io}(2-\B + 5*)(5(2-' (B + 5*)))
-
zco((20"^ (B
-
^
)^((2z)~^ (B
-
r)))
>
0
and hence
-ico(B''ö(B)) Lemma 5.3.16
ical
unitary
io}(BÖ(B'')}
implies that co is i-invariant. Let implementing T on ^ The
group
(no,(B)Qo.,H,,7i,,(B)Ü^) for all B G
-
D(ö).
Next
we use
7r(T/(5))Q which is valid for
/
G
-
+
t
^-^
>
0
.
t/to(0
exp{zY//(o}
be the
canon-
last condition then reads
(7r,X5*)Q,o,//o;7ra.(^*)aj
0
>
the relation
[/,,(/)7c^(5)Q^
5^(U). Moreover, T/(5)
(n^(B)ü^, H,,x(H^)n^(B)Q^)
+
-
G
f(-H^)7i,,(B}^^ /)((5)
for all
/
G
e^ and hence
^(5)*^., //c.z(-//.)7r(5)*Üo.)
>
0
104
States in
Quantum Statistical Mechanics
for all 5 G ^ and all in
Tif/X^)
=
9}ifo,
positive
and thus
x ^
^(tR). By Corollary 2.4.15, 7c,,(^)
is
strong*-dense
have established
we
OBSERVATION l
(^ao,//c.X(^roMna.) for all A e a^.o and all
positive
/ e
[7rf^(3J)'Qf^].
Next, define E
(^*a., //coZ(-/^..M*^.o)
+
0
>
^((R).
Then Q^^ is
cyclic and separating for E^'^l^E. Let A^, on 'fy. Extend
JE be the associated modular operator and antiunitary involution A to ,^ by defining A Then A is
=
A^'
.
nonnegative selfadjoint operator on ^y, and "^l^^^^o is E^^E^^,^ is a core for A^". By Standard results Takesaki theory we obtain a
since E'aJi^jQtu
(A^/-^Q,A^/'5ao) Since
=
(A^-^^^Q^, Ay-^-^^-ao)
=
(EB*EQ,,,EA*EQa,)
U,^(t)E
^r^ and f/,,(07^<:o(^)'^.o(0* t/,,(OO EU^(t] for all t, and hence
Since
A^
=
=
for
A'/^,
of the Tomita-
('F^O, ^*Q,,)
=
^o.(^)'
for
all
t,
we
then
find
=
t/(0^9JI,,^t/,,(0*-^aRco^ =
a core
is
AEU(o(t)E,
related
canonically
to
.
(E^,aE,^,o).
and hence A and H^^ commute
strongly.
it
follows
Thus
one
has
that a
Uco(t)EAE
joint spectral
representation,
je dE(s, A)
HO,=
A
,
where A
==
IR
x
[R and [R
=
=
/ e^'dE(8, 1)
,
JA
JA
IRu{-oo}.
Let
ä(//to, log A)
C
A be the support of "(-,
),
i.e., the joint spectrum of //^j and log A. OBSERVATION 2
a^(//a log A) To prove
this,
we
{(,;.) 6A;/<0}
c
apply Observation
l with
^(0
=
^~^~
.
on
A
=
BE where B G
^ca-
Therefore,
(BÜ,,, where
we
f-^/()
=
transform
H^e-^'^B^,,]
+
('5*^,,,
H^^e'^'^B^'Ü,,)
0
,
have used the fact that /4j and E commute strongly. The function ee"^" is real and anti-symmetric and hence has purely imaginary Fourier
/.
Let C E
SO^oj be the element defined by C*
=
j dtU^(t]B^U^(tYf(t]
.
Then
C
and hence
>
=
-
j dtU,,(t]BU,,(tYf(t]
,
105
KMS-States
C*Q,,
Using
-
//^ß-^-m^
the modular relation for A
(^ra^;,
=
-H^e-"'^B^^
.
obtain
we
//co^~^''^*a^)
Cüa,
,
-
(^5*^0^, c*n^)
=
(A^/^Cl^,, A^%Q,)
-
-(A^/2//,e-^^5Q, A^/^^Q,)
,
and hence
H^e-^'^BQ^)
(BQ^, As 9}ia;^co is
a core
for
A^/^,
(l^^'^H^e-^^^B^^, A^%Q^)
-
and //o; commutes
with
strongly
0
>
.
A''^^,
it follows
by
closure that
//,,e-^^(1]-A)>0. Observation 2 follows
a(H^, logA)
Define
immediately =
from this
d(Ha,, logA)nlR
inequality.
[R
x
.
OBSERVATION 3
ö-(//,logA)--ö-(//,, logA). Note that
o(H(^, log A)
=
Hence Observation 3 follows from the
(T(Ha)E, logA^).
identities.
JE A: JE
=
The first of these follows from
of e^^"^^ with SE orem
=
JR^T"
^E
JE H^ EJE
,
2.5.1
Proposition
which is established
(1) (2) (3)
I + E C
äs
in the
proof
of
(3)
->
(2)
of The
Then
co
Assume that there exists
I, where
Z
=
i:n(IR
x
R)
a
set Z C {R
x
!R, such that
,
{(,/l) E [Rx R; e;i<0} ä(//,, logA)ci:. ZC
In the
is
a
proof
(T, j5)-KMS
state for some
of Observation 4
a(H^,\ogl^]
Case L
=
ß
through (0,0) (otherwise {n p not be contained in
{(e,A)
G
E
-\- mq\ n,
U^\X
[0, CXD]
consider two
we
cr(/4j5logA). Thus,
servation 3, and hence /? + TTI^ E Z for all assumption (2) implies that p and q must m
<
E
cases
separately.
if p,q e a, then -p, q, E ä
would
by
Ob
by assumption (1). But then the same straight line passing form a lattice in [R^ which could
integers , belong to
Z}
O}).
.
m
It follows that
(7(//a;,log A)
itself is
con-
0 and straight line passing through (0, 0). If this line is vertical, then /f is trivially a ground state. If the line is not vertical, there exists ^ > 0 such that -ßs for all (e,/l) E (j(Haj, log A). But then
tained in
;i
.
1, the second from the commutation
3.2.61.
OBSERVATION 4.
co
/4; E
=
a
:=
^
=
^
and hence Case
(EO, -oo)
co
is
a
=
i-KMS state at value
e-ß"-
,
ß by
Theorem 5.3.10.
II.
Then
E
E
ö-(/4j, logA) ^ ö-(//a;, logA). ö-(/4j, log A). Assume that (E, }.)
there
exists
ö-(//cü, log A)
for
EQ E [R
some
<
such
that
0. Then
(3)
106
States in Quantum Statistical Mechanics
and
(2) imply
n(, 1)
that /l >
(fio, -oo)
+
=
0, and (e, /l)
(n +
-oo)
fio,
e
ö-(//oj, logA).
for all
G Z
ns-\- Q < 0 this contradicts
(2). Therefore, 8 is a ground state.
means
that H^o
>
We
are now
able to prove
0 and
co
(4)
>
0 for any
(1) of Theorem
^
But
then
(1)
N, and when
G
77
(g, 1)
5.3.22. The
G
implies
that
large ff(H(, log A).
This
n
is
so
proof is based
that
on
the
following formula: /
\\
/
älHn
V^
,
"
-yö-(//,
A
log
V
00.
(8)a.// ^
7=1
logA)
f:rf
^'^
J=l
which follows from the obvious facts that /;
H
=
n
0
Vl
0
.
-
0 l 0//,o (8) H (S)
-
TT^
W
.
.
.
O D
A'\,
,
J
Hence, if
7'=
satisfies condition
co
co
A'^
.
y=i
l
(4), i.e., ^"^ico satisfies condition (5) for
follows from Observation 2 that the /
all n, it
set \
n
U E^(^-'i^g^)
^=
>iV/'=i satisfies condition
^
-
^
7=1
/
of Observation 4. Conditions
(2) (1) and (3) of this observation are clearly satisfied, and hence Observation 4 implies that o; is a r-KMS state at value ß G [0,00]. In particular, co is t-invariant by Proposition 5.3.3 or Lemma 5.3.16. Finally, we prove that (5) ^ (1) if co is weakly G-clustering for an action a of a group G on ^ such that (y.g it We show in this case that the
=
a,y, for all ^ G G and t G R, and if co is a- invariant. of Observation 4 are fulfilled with
T/
requirements t
=
a(H^,log^)
.
Condition (3) is then trivial, while condition (2) follows from Observation 2. It remains to show condition (1). Let co, T, and a be the normal extensions of co, T, and a to 9JJco Theorems 4.3.22 and 4.3.23 then imply that a, restricted to Ey)l,E, =
TToX^)".
has the three
point düster property: inf B'eCo(aG(B)}
for all
Ä,B,C
e
l cb(AB'C]
-
(:b(AC)cb(B) V y V / \l
=
0
E^Rf^E. Hence it follows from Theorem 4.3.33 and the subsequent
remark that S
has the
-
ö-(//, log A)
-
a(H,E, logA^)
semigroup property z + zcs.
(Observation
3
even
implies
that Z is
a
group.) Thus
Observation 4, it is enough to show that (g, (e + s', oo) G Z. But these conditions mean that 8
Now
äs
E is
exists A,B
e
e
(7(H,,(^
-
E))
,
B'
-oo)
to show condition
S and
G
e(T(H,,E)
(g',;.)
G Z
(1) in implies
.
t-invariant, it follows from Proposition 3.2.40 that for ^co such that
any ^ > 0 there
KMS-States
A
=
(i
E)AE
-
G^(A] But
commutes with
äs a
(g
C
A^a, 7^ 0
,
(5,
-
E
^)
+
B
,
C7,(5)
,
the last condition
T
EBE
=
o,(U,,(Q)BU,,(g)''}
(e'
C
implies
(5, e'
-
5)
+
,
.
that
'
('-(5,
C
BQ.,o + 0
,
107
+
(5)
for all g e G. Hence it follows from Lemma 3.2.42 that
(Tu^,^(AU,,(g)BU,,(gYa>,, )
(s-i- s'
C
-
20,
+
s' +
20)
for all ^ e G. As (H E)A A, the latter vector lies in (D E)^^ for all g e G. Hence we can show that the vector is nonzero for some ^ G G, it follows that =
-
-
if
ö-(//,o(l As ö
this
arbitrary,
was
E))n(
-
+
8'
-
2^,
8
e' + 2^) / 0
+
.
implies
(8
',-oo)
+
62:
.
Assume, ad absurdum, that
AU,,(g)BU,,(gYü, for all ^
G. Then
AU,o(g)BU,,(gYE
U,,(g)BU,,(gY(^-E) by
the
assumption
BE
=
0
=
by
the definition of ". As
U,,(g)B(^
=
or
AU,(g)B
=
E)U,,(gY
-
=
0
B, it follows that
AU,,(g)BU,,(gY for all g,
0
=
0
=
0. Thus
(AQ,,,AUMBB''Ü,}=0. But the weak
G-clustering
of
oj
with respect to
|Mn,,||^||5*Q,,f But
äs
AQ.,^ 7^ 0 this implies 5*0,^
hence 5*
=
proof
it cxo"
0
of Theorem 5.3.22 is
imply
that
3 B'
and
.
=
for
EWl,E
sci:
complete.
We conclude this subsection with
a
and several comments and
discussion of various convergence of
examples concerning the existence States. First we derive a justification for a terminology "KMS state used in the definition of ground states and ceiling states.
properties KMS
=
and Theorem 4.3.22 next
0. But Q,o is separating assumption B^a^ / 0. Thus
0. This contradicts the
E +
and the
a
Proposition states on
5.3.23.
Let
(^, T)
be
a
^l such that co^ converges
lim
at
C"" -dynamical System, and
to a state CD in
o}r^(A)
=
the
{o^y} a net of weak'^-topology, i.e.,
(D(A)
a
for
all A e ^. Assume that co^ is
a
(T, ß^)-KMS state,
and that
\imß,
=
ß
where
ßy
G
[Ru{oc}.
108
States in
Quantum Statistical Mechanics
exists in R u { dz
cxo
Itfollows that PROOF.
If
Hence the
}
.
is
(D
\ ß < 00, corollary
we
is
(i:,ß]-KMS
a
may
state.
that
assume
^^ |<
for all
oo
by passing
a
to
a
subnet.
immediate consequence of the auto-correlation lower bounds in Theorem 5.3.15, since u, v H^ u \og(u/v) is lower semi-continuous. If ^ + 00, we may either assume that ß^ ^oo for all a, or 0 < jß.^ < + cx) for all a. In the first case an
=
=
-ia}(Ä''-ö(A))
lim-i(Do,(A-''ö(A)}
=
>
0
a
for all A e
D(ö).
In the second
-ico(A'^3(A))
case
note that
by
the auto-correlation lower bound
\im-ico.,(A^Ö(A))
=
y.
ins
>
^
^
G
D(d).
\Q^(u/v)
u
by convexity The
case
of
ß
The last
=
.x
i^
;c
oo
ßa
inequahty
=
log
Pa
iim
>
for all A
^ a;,(^*^)log(co(^*.^)/co,(^^*)) ((Dr^(A^A]
-
O^MÄ"]]
0
=
follows because
v((u/v] log(i//i;))
>
v(ulv
1)
-
u
:=
-
v
x.
is similar.
EXAMPLE 5.3.24. Adopt the assumptions of Examples 5.3.2 unique (T,jß)-KMS state over the CAR algebra ^(I)), for ß e [R, variant quasi free-state co/j with two-point function
o,ß(a'\f}a(g)] If there
are no nonzero
lim /j^
/
6
=
(g, e-/"'(1
+
(^ invariant under U,
a)/,(a*(/)a(ö))
=
e"/'^)-'/) =
and 5.3.20. The
is the
gauge-in-
.
e''^ tlien ,
(g, ^(-<^,0)/)
,
00
where EH is the
spectral family of //. topology to the unique ground state /, ^ G I) are invariant under Ut then
Thus the KMS states converge in the weak*constructed in Example 5.3.20. Note that if
cop(a\f]a(g)] Thus the co/j are trace-states variant vectors.
on
=
the CAR
(gJ]/2. subalgebra
over
the
subspace
of ^-in
The next
proposition, which is similar to the previous one, is sometimes proving existence of ground states and KMS states in C*-dynamical Systems which are thermodynamic limits. It also implies that if ^11 has an identity, the set of ^ such that (T, /5)-KMS states exist, for a given group T, is a useful for
closed subset of
[Ru{oo}.
Proposition 5.3.25. Let "^ be a C"" -algebra with identity D, and {T''},;>I a of strongly continuoiis one-parameter groiips of'^-automorphisms of Ul converging strongly to a one-parameter group T, i.e.,
seqiience
KMS-States
lim
each
077
^for
0
=
oo
>
for
||T;'(^)-t,(^)||
IR and A e ^. Assume that there exists
^ G
each
where
n,
109
{ßn}>i
C
[Ruit
lim
ß,
=
a
converges to
oo
(i^, ß^]-KMS state co a ß e ^^ {00}, i.e.,
ß.
n -^00
It
follows
KMS
that each weak'^-limit
State on
PROOF.
Let
^. In
point
particular there
^ be the generator of
and 3.1.28 there exists for each A G lim An //
=
A
CD
exists
of a
the sequence
(T, ß]-KMS
T" and d the generator of
D(d]
a
sequence
lim
,
00
^(An}
{An}f^>i
=
ö(A)
{(DU}
is
state on
a
(1,^8)-
^.
By Theorems 3.1.26
T.
in ^ such that
.
n^00
Let cüfj be a (T",^,J-KMS state of ^. Since ^ has an identity, E^i is compact in the weak*-topology, Theorem, 2.3.15. Thus there exists a subset {ca^}^ of {co,,};, such that a cxD and co,,.^ converges to some state co in the weak*-topology. We will show >
that
co
If
is
a
ß \<
(T,^)-KMS
state.
that
oo, we may assume
\ ß^j |<
oo
for all n, and
by the auto-correlation
lower bounds, Theorem 5.3.15,
(A* A
(CD
]
'/:"":( ^nA^n:,^nJ
Hence, by limiting
-/^co(^(5(^))
>
iim{co,J^;^J log (co,J^;^,J/co,J^XJ)}
>
co(^M)log(w(^*^)/co(^^*))
where the last step uses the lower semi-continuity of u^v If ß -{-oo for all a, H-CXO, we may either assume ß^^ =
In the first
=
wlog(w/i').
i-^
0 <
or
j5^
< H-CXD
for all
a.
case
-ico^(Alö(A,J)>Q for all a, and
essentially by
co
is
the
ground reasoning a
EXAMPLE 5.3.26. that
(f,,, 7L(a)
is
a
state
by limiting. In the second case c/j is proof of Proposition 5.3.23.
ground
a
state
used in the
Let ^ be
a
C*-algebra
with
identity,
faithful representation, and consider
a
co a
trace-state
sequence
//
over
51 such
// * G ^ such
that
ö(A)=
lim
/[//, A]
n -^00
exists for all .4 in
a dense *-subalgebra D(Ö)Q of Ul. Now wo ö Q and ö is closable by Proposition 3.2.26 and if the subspaces (z Ö}(D(Ö)Q) are dense in ^, the closure 5 of ^ is the generator of a group T of *-automorphisms of 5t by Corollary 3.2.57. If i';(A) e''""Ae~'^"", then T'/(^) T;(y4) uniformly for t in compacts, Theorem 3.1.28. =
-^
=
Now
(Dß^n(A}^co(e-^^"A)/w(e-ß^") is
a
(t",^)-KMS
state
on
51, since
110
States in
Quantum Statistical Mechanics
(OiUAB)
=
=
o4e-l>"''AB)/w(e^^"") a,(e-i>"'e"-Be-i^"'A)/co(e-^"-)
=
Wn(^"_,^(B]A)
for all
^,5 e 51, and it is in fact not hard to see that this establishes a one-to-one correspondence between (T'^,^)-KMS states and trace-states on ^. By Proposition 5.3.25, 51 has (T, jS)-KMS states for all j5 e (Ru {00}. Our fascination with KMS states
into
thinking
that such states
shows that this is not
throughout
this section
exist. We conclude with
always necessarily the
might mislead one an example which
case.
EXAMPLE 5.3.27.
Our aim is to exhibit some C*-dynamical Systems (51, T), where simple C*-algebras with identity, and there exists a t-KMS state at one and only one value ß 6 [Ru{cx)}. This state will be unique. Furthermore, this is possible both for finite and infinite ß, and hence by rescaling of T for any ß e Ru{oo}. For /7 2, 3, +00, let 0 be the C*-algebra generated by a sequence {Si}"^ j of the 51
are
==
.
.
.
,
isometries
(1) and if
S^S,= i^ /7
< + oo, assume
z-
l, 2,...
that
Y.
(2.n)
^*^'
=
^
/=!
If
/2
=
+00,
assume
that n
^SiS*
(2.(X))
<
1;
k=
1,2,....
/=!
can be proved (see l by elements S^, i
It
Notes and
=
a;
&n
H->
(9^^
Remarks) that if (9^^ is another C*-algebra generated satisfying (1) and (2. n), then there exists a *-isomorphism a(5'/) S'-. Hence & is uniquely determined up to *-iso(1) and (2. n), and by the arguments used in the uniqueness ,n
such that
=
morphism by relations proofs of CARs and CCRs, Theorems simple C*-algebra. Now for each t
Sl
=
5.2.5 and 5.2.8, it follows that each (9n is
a
i^(S^)=e^^Si
satisfies
(1) and (2. n), and hence T^ extends uniquely to a *-automorphism of (9 (This is the only point in the argument at which we use the uniqueness of the (9'. Alternatively one could regard each T; äs a representation of relations (1) and (2. n), and then form the direct sum representation ©^^^^T^- Trivially (@s^^T^s)(Si) f^e'' X (@^,^^^T,,)(5/) defines a*-automorphism of the C*-algebra 0^ generated by this representation, and the remaining arguments do not change. Of course, the simplicity of 0^ would need another proof.) Clearly ^ ^ T; is a strongly continuous one-parameter group, which is periodic with period 27ü. Hence n-
'^(^)=^r^'^'(^) defines
a
projection
of
norm one
from ^
onto the
fixed-point algebra 0^^^
for
T.
KMS-States
We next
this
analyze
It follows
fixed-point algebra.
immediately
from
111
(1)
and
that
(2.n)
(3)
S]Si *
and hence each element in the
öij^',
=
ij'=\,2,..,n
-algebra generated by {^S/}"^ j
is
linear combination
a
of elements of the form
S^^v where /x,
multiindices, e.g.,
v are
//
Define the
of /z
length
(/i, ...Jk)Ji
=
Sj^ Sj,
S^ (J\, ...,7^)
'
.
.
L(i^)
äs
Sj^
.
=
{1,2, ...,},
6
and
.
k. Then
T,(^A*)-e''(^(^')-^(^))V: and hence
^(S^S^,) It follows that
form
S^S;,
is
0l
where
just
L(//)
=
=
^i(^),i(v)'S'^'5'*
.
the closure of the linear combinations of elements of the
L(v).
But if L(i^)
=
L(v)
L(^]
=
=
L(V)
it follows from
(3)
that ^^
*-*//JV'-^M
^')/''
v'
^I^^V'
5
i.e., the set of S^S* with L(//) L(v) k form a set of ^' x ^ matrix units. It follows from (2. n) when n < -}-oo that these matrix units are linear combinations of n of the corresponding matrix units for L(ij) L(v) k-{- 1, and hence 0^ is a UHF-algebra, i.e., (9^ is generated by an increasing sequence {Syt}jr.>j of füll n^ x n^ matrix alge4- oo, a Variation of the argument above shows bras, all with the same unit. If =
=
=
=
=
that
01^
Hubert
^^(S>)
can
be
represented
space, then 0 D 0 H <8)
,
äs
(P^ is the =^^( (g) )
tracial state which is faithful if
Now, let
co
be
a
follows: If
state
on
n
<
(g) H 0 oo
,
and is
(P which is
co(Tt(s,s:))
is
C* -algebra
=
etc. It a
infinite-dimensional
an
on
(8)0-" follows that
character if
t-KMS state at
a
e''^(^(^')-^(^'))co(v:)
=
a
separable generated by 1, (9^ admits a unique
+
oo.
finite value
j5
G IR.
As
,
it follows from the i-invariance of co,
0 unless Proposition 5.3.3, that o}(Sf^S^,) that does admit not & (Note L(v). any tracial state because of relations (1) and (2. n) and hence the case ^ 0 is immediately excluded.) Now, Proposition 5.3.28 implies that the restriction of co to (9l is a trace, which is faithful, since & is simple. This shows that (0^, T) does not have any KMS states for finite values of ß,
L(}ji)
=
=
and if
< +
cxD we
have shown that
(4)
co(S,S:)
0 ifL(^) ^ L(v)). (where ö^^^ ß if, and only if, =
Hence
co
ö,^,n-^^^^^
=
is
unique,
if it exists. But
value
co(s,>s:,iiß(s,s;))
=
co(s,s:,s,>s:)
,
i.e.,
e-ß(^(^^^-^(^'^^co(S,^S:.S,s:)
=
CD(S,S;S,>S:,)
.
co
is
a
KMS state at
112
States in
Quantum Statistical Mechanics
It follows from
(3) and (4) by tedious arguments that tlie left-hand side (LHS) of this equation is nonzero if, and only if, the right-hand side (RHS) is nonzero, and in this case
I(/0+^(ri-^(vO+^(v) Furthermore, when this condition is fulfilled, LHS
=
e-''W")-^W)
RHs=("-7';' [/r^(^'), But
L(ß]+L(i.i']
etc. Hence it
for all /.i, v,
=
L(v)+L(V)
follows that the
has
/"'"''
if^(/')>i(v')
[/?-'-("'),
if
!(/()< i(v')
,
f^(''')>^v) ifL(//)
and hence
equation
L(I.I)
>
.
L(v') if,
is satisfied if and
and
only
only if, L(/0
^(v),
<
if
i.e., if and only if,
ß Now,
one
.
assume /? <
H-
oo
and let
oj
=
be
log
n
.
ground
a
on
(^'n. Then
t^oj(S,i,(Sj)}=e-^'cD(S,S^} has
a
bounded
analytic extension
to the upper
and hence
half-plane,
co(5/S*)
=
0.
But then n
com
=Y,^o(SiS;}
Q
=
/=!
which is
a
contradiction. To show nonexistence of
ceiling
states for all /?,
notes
one
that
i:^co(S;i,(S,))=e-'^CDm=e^^ does not have
If
choose
G
all /.
a bounded analytic extension to the lower half-plane. representation for {5/}J^ , such that there exists a unit vector O If co is the corresponding state, then
a
co(SS^]~!^^
^'XVv)-|o,
for/(^v-0,
(S,S;
=
n
=
+00,
R(Sf)'^
for
^)
otherwise.
It follows
immediately that co is a ground state. Conversely, it follows from the reasoning showing nonexistence of ground states for finite n that this is the only ground state. Condiision. The C*-dynamical System ((^^'/;,T), 2. 3, + 00 admits a (T,/i)KMS state if, and only if, ß +00- if // log /?, (ß +oc). The corresponding KMS state is unique. =
...,
=
=
=
5.3.2. The Set of KMS States The KMS condition
originated
characteristic of Gibbs
equilibriiim
states
in qiiantum Statistical mechanics and for finite Systems the condition
com-
äs a
KMS-States
pletely characterizes
the Gibbs states
113
(see Example 5.3.31). This coincidence
between KMS and Gibbs states appears to persist in many models after the thermodynamic limit. The ideal Fermi gas provides a specific example. Thus if
thermodynamic System is described by a C*-dynamical System (^, T) and one accepts the Gibbs formalism it is natural to interpret the set Kß of (T, ^)-KMS a
states
äs
the set of
fundamental
states at inverse
equilibrium
temperature ß. We will give
for this
interpretation later in this chapter and in chapter. If the model reflects the basic elements of physical reality one expects for high temperatures that Kß contains a unique element but at lower temperatures, i.e., larger ß, it should contain many elements corresponding to the various thermodynamic phases and their possible mixtures. Our next intention is to analyze the set Kß with this interpretation in mind. In particular it is of interest to characterize the elements of Kß corresponding to pure phases and to examine the decomposition of a general co E Kß in terms of these special more
reasons
the next
elements. It is evident that the set
follows
immediately
Kß is a convex subset of the state space E^ but it from the auto-correlation lower bounds in Theorem 5.3.15
that this set is closed in the subset of E^ and
convex
weak*-topology. consequently it is
Thus if ^ has the
convex
an
identity, Kß
is
a
closure of its extreme
points. These extremal (T, jS)-KMS states appear äs natural candidates for the description of pure thermodynamic phases and then the Separation of phases coincides with the barycentric decomposition of KMS states. In Order to analyze Kß it is necessary to study the set of KMS states over an associated ^*-dynamical System (9^1, T). We begin our analysis with a proposition which implies, among other things, that a nontrivial one-parameter group on an abelian algebra does not admit faithful KMS states. This indicates that the KMS condition is not an appropriate characterization of equilibrium for classical Systems (see Notes and Remarks). Proposition 5.3.28. Let ^ be a von Neumann algebra with a cyclic iinit vector Q, co the corresponding state, and T a a-weakly continuoiis one-para meter group of'-automorphisms of^, Let 9Jr be the fixed-point
^o,
If co
is
a
{y^
G
{^
G
971; T,(^)
9Jl; 0}(AB)
all
for
w(BA}for
=
o/co.
-
m\
particular m^m'
and
A
i-KMS state then it follows that
(,, In
=
//9Jl
is
abelian,
one
t G
(R)
and let
algebra of i:
=
be the centralizer
=
has T/
c
i.for
a)r
all
t e
U
.
all B G
931}
114
States in
Quantum Statistical Mechanics
If A e
PROOF.
9[R\ then
e"^^ and T/(^)
A
CD(BA) for all B e'^r, and
Conversely,
if
the
by
co(AB)
CD(Bit(A)) because
is
co
=
-
=
A. Hence
by
the i-KMS condition
a}(AB)
=
911^ in 93l one concludes that density co(BA) for all 5 E 911, then of
(jo(i:t(A)B)
=
(D(i^t(B]A]
(D(Ai:_t(B)]
-
=
A
eO^co-
0
r-invariant, Proposition 5.3.3. Hence, if FB^A is the KMS function couple B, A, we have
associated with the
FBA^-I) Now, if 5
ayJt,
e
by
of the
edge
5.3.7. Hence
FB^A
analytic function
FB^A(t),t^U.
have that
we
FB^A(Z) the
=
co(T_,(5M)
=
wedge
argument used in the
has
entire
proof of (3) =^ (1) of Proposition analytic extension in this case. But äs an entire determined by its restriction to U. It follows that
it is
an
FB,A(Z -i) for all Im
z
z
<
FB^A(Z)
=
G C, i.e., FB^ A is periodic with period /. As FB^ A is bounded in the strip l < 0, it follows that FB^A is bounded, and hence FB^A is constant by Liouville's
theorem, i.e.,
(5*n, T,(A)Q) Therefore
äs
911^^
=
is dense in
FB,A(t)
=
for all
t e
A G
U. But Q is
9JJ^,
and ^(^
(5*0,
=
A
fi)
.
,
T:f(A)ü i.e.,
FS,A(O)
for 9}!
separating
Aü
=
by Corollary 5.3.9, and therefore Tf(A} A, proposition follows from the =
91!^. The last Statement of the
=
fact that
9}ln9JI'
The next result is
Proposition State on
a
key lemma
5.3.29. the
co, (Jt
Let ^ be
corresponding
C
in the
a von
e,^
.
analysis
Neumann
of
Kß.
algebra
with
afaithful normal
modular groiip, and cp any other normal
state
m. The foHovving Statements
(1) (2)
cp is
a
are
eqiiivalent.
a-KMS state,
There exists
a
positive operator
(p(A) for If these
=
T
afßliated
co(r^/-.4r^/-)
all Aem.
Statements
are
triie, then T is
unique.
with 9Jln9Jt' such that
KMS-States
In particular factor.
PROOF.
We may
co
is the
assume
unique a-KMS
the existence of
(ß(A]
a
state on
5DI
separating
and
(1)
=>
(9Jl, ß), Definition
a
(p < IcD
By Theorem 2.3.19, there then exists
(p(A) As (p is cr-KMS it is a-invariant
a\\A,Be^ and
a
=
r G [R.
/l
931 is
a
vector Q such that
and modular operator
0 such that
>
.
positive T'
(Q.AT^Q,)
G
W such that
.
5.3.3. It follows that
-
(Q, (7_,(y^*5)rQ)
=
(p(c^-^(A^B))
=
(p(A*B)
=
(AÜ., T'BQ.)
,
hence
^itj^f^-it for all
conjugation
by Proposition
(^a,A''rA-^'^5Q)
for
cyclic
only if,
2.5.10.
Assume first that there exists
(2).
and
(Q, AD.)
=
for all A e'^. Let J and A be the modular
associated with
if,
115
j^f
^
Now, define T
=
JT'J
.
Then rO
=
JT'Ü.
=
JT'A-^^~D
-jA-'/^ro
rn
-
,
where the third have
A'
G
follows from the strong commutation of T' and A that we The fourth stems from the fact that JA~^^^A'Q. ^'*Q for
equality proved previously. 501', Proposition 2.5.11.
It follows that
(p(A)
=
(Q, ATÜ)
=
a}(AT}
and it remains to prove that T G W. To this end, KMS condition to compute
0}(ATBC)
=
,
assume
A,B,C
e
'^a and
use
the
co((Ti(BC)AT}
=
(p((yi(BC)A)
=
(p(c7i(B}ai(C)A)
=
cp(ai(C)AB}
=
w((Ji(C]ABT]
=
0}(ABTC)
,
It follows that B and T commute and hence T eW.
It (f> is not bounded
(CD
(p)/2.
by
a
As p is a T-KMS part of the proof that there exist p
=
-}-
positive multiple of state and
faithful
on
co,
we
form the
new
state
5DZ, it follows from the first
positive operators ^1,72
G
9J^n9[)Z' such that
States in
116
Quantum Statistical Mechanics
co(A]=p(AT,)=p(Tl''^ATl'~] cp(A]
p(AT2]=p(T^J~AT^J^]
=
.
' faithful, the kernel of T\ must be zero, and hence Tf exists äs 72 Tf^ is possibly unbounded, operator affiliated with ^r\W Thus T
Since
is
co
=
.
a a
positive, positive
operator affiliated with äl^JnW. But then
co(r^/2^r^/2)
=
co(r;/^r^/^^r^/^r-
=
p(Tl^'~ATlf~)
cp(A}
=
for all ^ e 93t.
(2)
(1). Approximating
=^
T
strongly by
bounded elements in9Jln9[R'we may
(p(AB)
The
=
co(ABT)
=
CD(ATB]
=
w((7i(B)AT)
increasing
an
(p((Ti(B}A)
=
positive,
.
of T follows because T and A commute
uniqueness
sequence of
that T is bounded. But then, if ^,5G9JJt
assume
strongly by Proposition
5.3.28, and hence
T^/2^ where ^ is the natural But the
representative
hence T
G
cone
A^/^r^/^Qe.^,
associated with the
vector for cp in this
is
(9[Rn9}r)_^
=
unique. proposition
The last Statement of the
We
in
is
cone
pair {9Ji,Q} by Proposition 2.5.26. is unique, by Theorem 2.5.31, and
now
trivial.
position to establish the principal affine properties of the set C*-dynamical System (^,T). We will assume that UI possesses an identity but this does not lead to any essential loss of generality. If "^t does not have an identity, one can introduce "iH CH + '^K and extend T to ^ by are now
a
of KMS States of
a
=
the definition
(a,^)
T;
There is over
^
one-to-one
a
for which
G
^^T,((o^,^))
(a,T,(^))G^.
=
correspondence between states co over "ill and states cb 1. This correspondence is given by extension and
||a) s)i||
==
restriction, i.e.,
a)((a,.4)) Moreover, the
proof
äs
of
^^
CD +
=
Proposition
=-
(T,j5)-KMS states of (^,T) dynamical System (^, T) with ||a)|s^|| =
an
identity
Theorem 5.3.30.
has
an
identity.
Let For
It follows that:
ß
+
co(v4)
^^ if follows by using 5.3.3 that this
tween the
the existence of
a
gives
a
approximate identity äs in one-to-one correspondence be
and the 1. The
.
an
(iJ)-KMS
states of the C*-
principal advantage in assuming weak*-topology.
is that E>^i is compact in the
(^, T) e U let
be
Kß
a
G^-dynamical System and assume that set ofi-KMS states at vahie ß,
be the
^
KMS-States
(1) (2) (3) (4)
Kß Kß co
is
convex
is
a
E
and
weak'^-compact,
Simplex is
Kß
an
extremal point
Let (jL)\ and CD^ be extremal
equal or disjoint. If 0} Kß^ the unique
(5)
117
identical
maximal
the central
to
of Kß if,
and
only if,
points of Kß, then
co
is
afactor
coi and 0)2
^f^
Kß corresponding corresponding t o co.
measure on
measure
state.
either
to co is
PROOF. If ^ ^ 0, we may assume ß l by rescaling T. Moreover, we already proved (1) in the introductory remarks of this subsection. R + K-i is just the set of positive functionals on ^ which satisfy the T(2) KMS condition. If co, (^ e (R + Ä^_i define p (D + (p. Since co < p.cp < p,ii follows from Theorem 2.3.19 that co and (p and Tip-nonnal, and Corollary 5.3.4 and Proposition 5.3.29 imply that there exist positive operators Ti,T2 G 71^(51) n np(^)' =
=
,
such that
co(A)
p(A(T,)
=
(p(A)=p(AT2)
,
for v4 e
51, where p is the normal extension of p to Tip (5l) As 3p Tip (51) n 7rp(5I)' abelian, the greatest lowest bound TI A T2 of TI and T2 exists in 3p- Define =
.
is
((jo/\(p)(A) for ^ G 5t. Then KM S T G
co
A (/) is
(3p)+
KMS
a
positive functional
T
then
< co, T < cp,
T
< p
5.3.29. If
T
is
and there exists
a a
such that
r, TI
äs
p(A(Ti/\T2))
positive functional by Proposition
such that
T(A) But
=
,
r2
G
=
p(AT]
.
it follows that
3p
T <TI
T
,
,
and hence r <
Thus [R
+
T
< co A (/),
and
so
co
A (^ is a
^_i. It follows that [R + Är_i is
(3) (p G IR
From +
Proposition
A^-i which
are
a
is
one
G
to one. Hence
co
TS
.
5.3.29 it follows that if
dominated
7Caj(5l)'n7ra;(5I)
A
unique greatest lower bound of lattice, and K-i is a simplex
by
(p(A) where T
Ji
and 0
co are
=
co
G
co
and (p in
K-\ is given, then the
of the form
cb(AT)
,
1], and the correspondence between is extremal in K-i if, and only if, < T <
7rc.(5I)"n7i^(5l)'-C1]
cp and T
.
As coi and C02 are factor states they are either quasi-equivalent or disjoint by (4) proof of Proposition 2.4.27. (Compare also Lemma 4.2.8). But if they are quasiequivalent they are equal by Proposition 5.3.29. Let ju^ be the unique maximal measure on K_} with barycenter (5) co, which exists by Theorem 4.1.15. Let //c be the central measure of co, i.e., is the or //^ thogonal measure mMc^(E^) corresponding to 71(^(51) n 7Coj(5I)'. If /z G Mco(K-\} is a measure of finite support, then /i l, and since X^/^'^w, where /l/ > 0 and J]/ A the
=
=
States in
118
/}(^)
^(/.i^)
=:
Quantum Statistical Mechanics
=
cü
it follows that co/
sitive operators T/ G
<
7ioj(^ )n7rf,j(^)'
(ßi(A)
co(ATi)
=
co/1/. By Proposition 5.3.29,
there exist po
such that
(Q,7r,,(^)rA.)
=
.
Applying Lemma 4.1.26 and approximating T/ by a linear combination of spectral projections it is now simple to show that /( ^ /i^ i^ the ordering of measures in Mf,j(E^i). As /i^, can be approximated by finite support measures in M(^(Ar_i), it follows that A% ^ /^c
On the other
in
M(,)(Eyi)
hand, Lemma 4.1.26 implies that i^i^ is
a
limit of finite support
measures
of the form /f
=
^ }.iö,o,
,
/
where
a)^(A) and P/
are
in
projections
^-1, and thus
co/ G
71(^(^0
^
co(ÄPi)
=
Ti^^jC^I)'-
But
Proposition 5.3.29 then implies Mf,j(Är_i). Hence
that
/f ^ /i^, since /.f^ is maximal in
A^c ^ A^A:
and
we
conclude from Lemma 4.1.4 that MC
The is
just
trivial
remaining
case,
^
=
0,
the set of trace-states
dynamics
it
=
can
on
be deduced from the above
since
convex
T
are
just
by remarking
the KMS states
on
that KQ
5.1 for the
i.
is then
an
strongly continuous (see Example 3.2.36). If one 50^ which satisfy the T-KMS condition Systems
^^K
^ and these
Theorem 5.3.30 does not have
(SJt, T)
=
obvioiis
analogue
continiious if, and defines
Kß äs the set ß then Kß
for W*-dynamical only if, it is normof normal states of
at value
subset of the normal states but it is not
is a weakly closed necessarily compact nor does it
always have extremal points. For example, if the centre of 9J1 has no minimal projections then Kß has no extremal points. The simplex property (2) is replaced by the property that if co G Kß and p is a state such that p < }.co for some 1 > 0 then p G Kß if and only if there is positive operator T G 9Jln9}l^ such that p(A) CD(AT) for all A e'^. Properties (3) and (4) remain true whilst (5) no longer makes sense. Theorem 5.3.30 has many interesting implications especially for the tentative physical interpretation of Kß äs the set of equilibrium states, of the System (^, T), at inverse temperature ß, If Kß consists of one, and only one, state co, this state is automatically a factor state. Thus if there is a unique equilibrium state, it is a factor state. This is of particular interest because we have already established that factor states can often be characterized by düster properties which reflect the absence of long-range correlations, or the absence of large fluctuations for the values of space-averaged observables. In Section 2.6 we elaborated these characterizations for quasi-local algebras and in Section 4.3 =
KMS-States
for invariant states
(see
119
in
particular, Example 4.3.24 and the discussion in the 4.3). But the absence of correlations or large fluctuations is typical of pure thermodynamic phases and these various points all indicate that pure phases should correspond to factor states and more precisely to extremal KMS states. But if one adopts this interpretation, the Simplex property of Kß has a clear significance. Under suitable separability conditions, e.g., ^ separable, the unique maximal measure /^^ on Kß with barycenter CD ^ Kß is concentrated on the extreme points S(Kß) of Kß. Thus co is represented in a unique manner äs a convex superposition of extremal KMS states. This decomposition then corresponds to the physical Separation of an equilibrium state into pure ther modynamic phases. The symmetry, or lack of symmetry, of these phases is then automatically determined and phenomena of broken symmetry, äs discussed in Section 4.3.4, occur if 3a; is not pointwise invariant under the corresponding Notes and Remarks to Section
symmetry group. Theorem 5.2.30 also has interest in the
analysis of concrete models. In such essentially reverses the foregoing reasoning. For example, if one can show that a particular model predicts a unique (r, ß)- KMS state, then this state is a factor state and has good düster properties. In particular this demonstrates that the equilibrium states of the ideal Fermi gas described in Theorem 5.2.24 are factor states. This could also be deduced by direct construction of the corresponding representations äs in Example 5.2.20 but in more complex models, such äs the quantum spin Systems discussed in Chapter 6, the direct construction is not possible and the abstract proof of factoriality and düster properties is particularly convenient. On the other hand, for finite Systems, one can again reverse the line of reasoning and deduce uniqueness from factoriality. applications
one
EXAMPLE 5.3.31.
dimensional space
Let ^
^.
==
Eor H
M
be the
H* ^ M
of
algebra define
T,(A]=e""Ae-'"
T
n x n
matrices
acting
on
the
/?-
by
,
then the Gibbs state
o}ß(A] is
(T, ^)-KMS
a
is
((0
a
+
second
state but it
is also the
(T,j5)-KMS
coß)/2f^"(Kß).
Tr,>,,(g~^^^) =
state
Trö^X^"^"") unique and
Thus the mixed state
such state for the
co^cDß,
(co4-co^)/2
then is not
following reason. (co -^ ojß)/2 e Kß a
If
oj
but
factor state. But all
type-I factor states and this gives a contradiction. If ^ ^'tf () and is infinite dimensional, a similar argument yields uniqueness. It is, however, CD + 51 of 51 obtained by necessary to argue with the aid of the extension 51 adjoining an identity. states over
M,i
are
=
==
Theorem 5.3.30 estabhshes that the 0}
unique decomposition of a KMS state decomposition and we this should be interpreted äs the Separation of co into pure
into extremal KMS states coincides with the central
have
argued
that
States in
120
Quantum Statistical Mechanics
thermodynamic phases. But there are other possible natural definitions of a pure phase and one might well suspect that purity corresponds to ergodicity among the T-invariant states. Therefore, we next study conditions under which the KMS decomposition coincides with the ergodic decomposition of co re lative to T. Once again a condition of asymptotic abelianness is of crucial importance. C'^-dynamical System, assiime that ^ has an KMS states at value ß if ß e ff^\{0}, and ofiidentity Kß KQ the set of i-invariant trace-states on ^. Take CD G Kß, and let ^(^ G Mco(Kß) be the unique maximal measure mth barycenter CD. Theorem 5.3.32
D
be the
follomng
conditions
There exists
(1)
a
a
set
,
The
be
(^, T)
Let
and let
are
equivalent:
unique maximal
measure
^E^^K
v\^here
Ef^
{^, co}
(2)
is
\veakly
asymptotic abelian in
lim
^/
l
{^, co}
In
dt
on
^. in the
mean
cü(^[T,(5), C]D)
sense
that
0
=
J^
> G ^.
forallA.B, C,
(3)
ö
and
,
denotes the i-invariant states
T -S^(X)
MCO('^J),
/.i^ G
is R-central,
particular an extremal if, and only if
invariant
(T, ß)-KMS
state
co
is extremal in
variant
^/
lim / ->S>00 i
for
all
A, 5, C,
PROOF.
Let 3[R
Z) G
=
^,
O
where
71^0(3.1)",
dt
o,(A\-L,(B), C]D)
=
0
J^
ß
and let
e R.
co
and f denote the normal extensions of
co
and
T
^.Ifß^ 0, then co satisfies the f-KMS condition at ß by Corollary 5.3.4 and hence ü^o is separating for ^M by Corollary 5.3.9. If co is a trace-state, co is a tracestate on 50t, and Q^^ is again separating for 931 by a simple application of Lemma 5.3.8. In any case [^Jl'Qw] 1 by Proposition 2.5.3, and the Kovacs-Szücs theorem, Proposition 4.3.8, implies the existence of a unique normal G-invariant projection M from 9J1 to 9J!^ where 9JI^ {^ G 5)1; f,(^) A for all t G U}. Proposition 4.3.8 and Example 4.3.5 imply that to
=
-=
M(A}Ü,
=
lim
=
i
T-s^oo
y
T-S-^oc
T
ib
/
dt
U,,(t)AQ,,
dt
Tt(A)^co
j^
/^
/
;
-
S^ Js ^
for all A
G
9Jl and in
But then
particular
the limit exists.
KMS-States
121
M(A)B'^a,=B'M(A}Üa^ /"^
l
^r^i^ T-S-^
00
^'^r~^^ J^ r^
l
^r^i^ TS^oo 501',
for all 5' e
and since ihQ nei T
7^-^/Js
dt^t(A]B'^^
1^
l
S
-
dtl,(A)^^
l
(T
^
-
f^dt it(A}
S)
is
uniformly
boun-
ded, it follows that
M(^)=
-^/
lim
7
O r>oo
^
dti,(A)
Js
exists in the strong operator topology. This ensures that the limit in condition (2) of the theorem always exists, and äs SÜl^Qß, is dense in ^ it follows from Definition 4.3.6 and Theorem 2.3.19 that conditions
conditions
are
equivalent
'm^ If J and A
{5m, Q},
are
one
that
(3)
are
equivalent,
and in fact both
the modular
c
mr^m'
conjugation
(*)
.
and the modular operator associated with
has
t/^W/A^/^^Qo; for ^ G
and
(2)
to
-
Uo,(t)A''Qcü
f,(^)*nco -^A^/^t/o^W^Q^
=
JA^'^^ it follows 9[R, and from the uniqueness of the polar decomposition 5* JUco(t) for all / e IR. It follows that (*) is equivalent to =
Uaj(t)J
=
'^^
=
5Ul'nC/^([R)'
J(mr^Uo,(R)')J m^m' j(mr^m')j
-
C
^
=
By Proposition 4.3.3 this is, however, equivalent measure /i^, which is subcentral, i.e.,
to
.
Mfo(E^) containing
a
unique
maximal
)" ^ MC
where /^^ is the central measure. Now, if j5 since trivially I^K ^ ^E: it follows that
5
7^ 0,
then ^^^
/^^
t>y
Theorem 5.3.30 and
^^K= ^^E If
ß
=
0^
note that for any
projection
" G 50l
Ae^^ is
a
T-invariant trace
the extremal
on
we
have that
(Ü,,,AEÜa,)
31, and it follows
decomposition coincides
^
äs
with the
in the
proof of Theorem 5.3.30 that decomposition into invariant traces,
i.e., ^K
Conversely
=
^E
'
Theorem 5.3.30 implies that i^f^ () fi^ f^r j5 7^ 0, while fif, -< fi^ foT ß since //^^ is dominated by the measure corresponding to the tracial decomposition, and the latter measure is equal to /i^ by Theorem 5.3.30 applied in the case -c i. The last Statement of the theorem follows from the fact that co G is ex =
=
=
^(Kß)
tremal invariant
if, and only if, f^E^
I^K^^co
'
States in
122
Quantum Statistical Mechanics
In the
Remark 1.
course
and condition
of the above
proof it
was
established that condition
for any i-invariant state co such that Q^o is version of Theorem 4.3.14 in the case that
äquivalent (3) (2) separating for 7i(jj(^)" This gives a [R. QCÜ is separating and G are
.
=
An immediate consequence of Theorem 5.3.32 is the following concerning the set Kß of KMS states at value ß e U. (We
Remark 2.
global adopt the
Statement
convention that ^o is the set of i-invariant
The following conditions
We
equivalent for
are
each
aface E^^, Kß (Ul, co) is weakly asymptotically abelian (^, co) is U-centralfor all CD e Kß,
(1) (2) (3)
is
(10
fixed ß
^R,
in
have to show that
only
traces.)
each
For
CD
G
is
(1)
equivalent
there
Kß
exists
in
mean
for
all
CD
Kß,
^
to a
iinique
maximal
measiire
^E^M,,(E^^],and /^
=
fe
where /ij^ is the iinique maximal
measiire
in
Kß is a simplex, this is again equivalent to (l"} For each CD E Kß and each maximal measure
Mco(Kß),
Since
^i(Kß}
^
l
/.( G
Ma)(E^^
one
has
.
((l'} clearly implies (l'O but (l"} and the simplex property ofKß imply that each pair of maximal measures in MCO(^S?I) must coincide.) The equivalence between (1) and (l'^) is a general characterization of closed faces in compact convex sets, and is proved äs follows. (1) => (1^^). If ^iis any measure in Mco(E^i), there exists a net /i^ G Mco(E^^) of measures with finite support converging in the weak*-topology to /i, by Pro l by the facial property of Kß, and hence position 4.1.1. But i-i^(Kß) =
^i(Kß} (f^
=
l.
^
Assume that
(1).
e
CD
Kß,
and that n
CD
=
y. ^i^i i
is
a convex
Let /i G
decomposition
Mf,;('Jj)
be
a
of
l
with co/ G E^, We have to show that co/ such that
CD
maximal
=
G
Kß.
measure
n
^ liöcoi z
By A^/
the Cartier
eM,o.(^s2i)
-
Fell
-
such that
-
-< M
.
l
Meyer theorem, Proposition 4.2.1, there exist
measures
123
KMS-States
^
^E^'-/^^
/= l
But
f^(Kß)
äs
by assumption (l")
l
=
it follows that
^i(Kß)
l for /
=
=
l
,
.
.
.
,
and hence
(Di Oi
It follows that
is
Kß
Remark 3. In the
face in
a
we use
(1)
are
^
(2).
we saw
IR, but
jS equivalent: G
Kß is a face in E^, na}(^) is abelian for
(1) (2) (Here
conditions
e
Kß
E^.
remark
preceding
circumstances when
general follomng
dLi,(o}']o}' ll d^i((D'](D'
=
all
CD
that
Kß
is
a
face in
is very seldom
Kß
a
and let T be
eKß
co
under
quite
Kß
^
the convention that ^o is the set of all trace LQi
^"^
face in E<^, i.e., the
a
positive
states.)
element in
nco(^)'.
The po
sitive linear functional
O}T(A) is dominated
implies T G
that
=
(T^co, na,(A)Q.o,}
positive multiple cor/||<^r|| is in Kß.
by
a
7Ccü(^)'n noj(^')",
and
we
have
^ G 21
,
of co, by Theorem 2.3.19, and condition (1) It follows from Proposition 5.3.29 that
proved
that
nco(^)' Cn^(^)'r^n^(^Y Since 71^; (M)" is
anti-isomorphic
to
7iö,(^) is abelian. (1). If co G Ä^^ is such
,
na)(^)' by
.
the Tomita-Takesaki
theorem, it
follows that
(2)
=^
n^(S&Y
=
that
n^(^)'
and hence all states dominated
7roj(9I) =
is abehan,
one
has that
7Ce,(2I)''n7r^(2X)' of
CD are (T, j?)-KMS states by Proposition Kß is a face in E<^. face Theorem 5.3.37. of "31, We shall see later that KQQ always is a Next we examine various relationships between automorphisms and KMS states. In physical applications the automorphisms correspond to symmetry transformations and thus the following result gives information conceming symmetry properties of equilibrium states. In Proposition 5.3.29 we proved that if er is a one-parameter group of -automorphisms of a von Neumann algebra 5[R, then the set of er- KM S states is affinely isomorphic with the set of normal states on a hereditary subalgebra of the Center of 50l. This implies various relations between automorphisms and
Theorem 2.3.19 and
*
KMS states.
by
a
multiple
5.3.29. It follows that
124
States in
5.3.33.
PropositioE normal State
be
Quantum Statistical Mechanics
9[R
on
(9JI, er) be a W-dynamical system, and co a faithfiil satisfying the (a, ß) -KMS condition,for some ß ^ R. Let a Lei
'""-aiitomorphism 6)/9JL Itfollows that: a
(1) (2) (3) (4)
co
o
is
oc
If IfoLOt If ocfft
cDo oc
0} o a
PROOF.
=
WC
co o a
is
co
and
=
CD o a
o^|sjr)jn9jr
D^,
for
cr^a
all t G R.
(a, ß)-KMS.
is
^^ ^^^^
identity
and
ß ^ 0,
then
the
pair
(D.
^,^G9JI, then by applying Proposition 5.3.7 a function F in Ch(T)ß), analytic in D/j such that
to
obtain
F(t)
=
(D(oc(A)(Jt(c^(B)))
=
0)0
F(t^iß)
=
(D((Jt((y-(B]]c^(A])
==
co o
(cT^aa, ß)-KMS
a
follows from the
(2)
eU then t G
9Jt.
state on
0 then acr^
ß ^
(Jtdfor all t <^t^ for all
If
(1)
oc(A),a(B),
Hence
=
=
(a~^aa,ß)-KMS
a =
by
state
the
of the
uniqueness
a(Ä(GC-^(Jta)(B)) a((a~'fj,a)(5)/4)
same
,
.
proposition.
j5-KMS
group associated with co, The
5.3.10, and (1).
orem
(3)
is
(4)
It follows from
an
immediate consequence of (1).
(3)
and
Proposition
(cooa)(^) for all A G SOI, where T is
a
=
5.3.29 that
co(r^/2^r^/')
positive operator affiliated with 3
^
50i
n
SOf'. In parti-
cular
co(^) for ^ G
-
CD(T^'~AT^'-]
3- But if 3 C(K] is the Gelfand representation of 3, there exists by the Riesz representation theorem a unique probability Radon measure /z on the compact =
Hausdorff space K such that
co(^)
-
/
d^i(x]A(x)
,
JK
where ^
H^^
is the Gelfand transform. But then
l d^i(x]A(x]= JK[ d^i(x}f(x}A(x)
JK for all ^
G
3, and hence
JK
f
=
l
by
the
uniqueness
Cü o (X
The Statements
Systems
(^, T) by
(l)-(3)
the
=
CD
of
same
co o a
broken
by
.
Proposition 5.3.33 are also reasoning. If a is a symmetry
true for
C*-dynamical
of the System, i.e., a *is a (T, jß)-KMS state
of 51 which commutes with T, and if co a (T, ß)-KMS state. If co o a 7^ co the symmetry is said to be The following theorem states that under some apparently
automorphism then
of j^i. Thus
is also co.
circumstances symmetries
general
are
not
broken.
KMS-States
Theorem
125
5.3.33A
(Fannes-Vanheuverzwijn-Verbeure). Let (^.i]be a C^-dynamical System, d the generator O/T, co ß (i:^ß)-KMS state of^ mth ß ^ R and oc a *-automorphism of^ such that ar/ r^a/ör all ^ G R. Assiime there exists a sequence of unitaries Un G D(d) such that
a(^)
lim
=
>
for
UnAU;^
00
all A e"^ and
sup||5(^,)ll
.
n
Itfollows
that CD o OC
i.e., the symmetry PROOF.
trivial
Replacing
so we
is not broken
a
T^
by
OBSERVATION 1.
-c-spectrum 0 <
a^(A)
^
-ißo}(A''ö(A})
=
be
[a,b]
follows -
CO
,
CD.
we
may
assume
ß
>
afinite
interval in [R
=
[R and
a)(A''A) \og(co(A*A)/o}(AA*))
<
ß(b
-
assume
a)co(^*^)
inequality is the auto-correlation lower bound inequality note that the spectral restriction on
For the second
case
^
=
0 is
that the
that
The first
PROOF.
Q. But the
>
Let I
C /. It
by
if necessary 0.
T_^
assume
may
=
.
of Theorem 5.3.15. A
implies
that the
measure
J/i^(l) introduced
implies
prior
to
-
Proposition
(n,,(A}Q,,,dE(l)n,,(A)Ü^) 5.3.14 is
supported
on
-ißcD(A''ö(A))
-
proposition
(^(A^'A) log(co(^*yl)/oj(^yi*))
=
ß(7i,,(A)Üa,,H,,7i,,(A)ü,,)
=
-ß
<
o}(A''A)\og(co(A'^A)/(D(AA'^))
-aß
l
Ja
dn,().]
ß(b -a)
j
-
Ja
=
-
f'd^M^.- ['d^^Wlog( f'd^^W/ f'd^,(A}e^^-
Ja
\
dn,(X)\og
dß,,().)
=
OBSERVATION 2.
There exists
a
o}(AA'') all Ae^.
Ja
( l d^L,().)/ l \ Ja
Ja
l
Ja
ß(b
-
a)w(A'A)
Ja
for
/. Therefore this
that
C >ö such that < C
oj((x(AA''})
Ja
.
d^i,(Ä)/^
126
States in
PROOF.
Set K
sequence
h
Quantum Statistical Mechanics
||(5([/;i) ||
sup
=
If ^ G ^ has bounded i-spectrum
.
of
positive C^-functions length at most one, such
interval of
on
[R
that
=
we can
find
a
finite
[R, each with compact support in
Y^hl
l in
=
a
neighborhood
of
an
(T-^(A).
Define
An
=
/ dv,(-A}
=
j dthn(-t]lt(A]
.
Then
(AA-^) For each
n
=
such that A
Y.
^^A(-^^}hnW'
-
;^co(^O
^ 0 we now apply the auto-correlation following inequalities:
.
lower bound to UfnA^
to obtain the first of the
CD(A:An) log(oj(AnA:)/Oj(U,nAnA:U:^))
+
ißoj(A:U:^ö(U^n)An)
CD(A:An) log((ß(A:An)/0,(U,nAnA:U:^)) + ißo4A:U:^ö(U,An)) ißco(A:ö(A)) co(A:An)log(oj(A:An)/co(A,,A:))
=
-
-
-ißw(Alö(An))
<
<ß(D(AlAn) The second
inequality
from the
CD(A^,An)\Og(co(A'^^An)/Co(AnA-;-^)}
.
follows from Observation 1. But
ißcD(A:u:^ö(U,n)An)\ so
-
<
ß \\ö(Un,)\\co(A:An)
<
ß Kco(A:An)
previous inequality
log(o4A:A)/o4U,AA:u:))
ß(K+l)
<
or
(a(A:A) where C
=
eiip(ß(K
+
1)).
Now
<
taking
W(A;,A) But
äs
[a, T;]
=
0, a(v4,;)
=
C(Ü(U,AA:U:,)
((^(A]]^^
<
so
(D(A'-A]
the limit
^
cco(oi(AA;))
adding <
m
these
C(D(ci(AA'])
oo
obtains
one
.
inequalities
over n one
finds
.
Since the elements with bounded i-spectrum are dense in 91, by Lemma 3.2.39 inequality extends to all A G 9t, and Observation 2 follows.
(4),
this
END
OF PROOF OF
(T, ^)-KMS
States
THEOREM 5. 3. 33 A: But then
It is sufficient to prove
is
extremal
co
(T, jß)-KMS
o
a
=
co
for extremal
By Observation 2 and Theorem 2.3.19, co is a normal state in the representation defined by co o a. But äs co o a is a factor state, by Theorem 5.3.30 (3), it follows from Proposition 5.3.29 (or the Simplex property of Kß) that co co o oc. This completes the proof of the co.
co o a
an
state.
=
theorem.
Any normal faithful state of a von Neumann algebra 5[R is a KMS state for a unique automorphism group of 50t, by Theorem 5.3.10. The converse is not true however; an automorphism group does not necessarily have KMS states. Proposition 5.3.28 implies, for example, that if T is an automorphism group
KMS-States
which allows
a
faithful KMS state, then T leaves the center of 9K on T is not sufficient however, by the
fixed. Even this condition
127
pointwise following
theorem, which states that any two groups which allow faithful KMS states are related by an inner cocycle. This theorem is a partial restatement of Theorem 2.7.16.
(Connes). Lei CD and (p be faithful normal states on a von algebra 50l, and let a^ and a^ be the corresponding modular groups. Itfollows that there exists a strongly continuous one-parameter famüy t^-^Tf of unitaries in 5IR such that
Theorem 5.3.34 Neumann
for
all A G 931, and all ^,
PROOF.
Define
a
^ G
af(A)
=
r,<(^)r;
r/+5
=
Yta^(Ts)
,
,
IR.
faithful normal state p
on
SOI (g) MZ
by
Kte t))4('"(^")+-(^-))Let (jf be the modular group associated with p. As
((\
OW^n
l
AU\\
((An An\(\
^
,,
^U OJU. ^22Jj=2"(^")=''iU ^22JU it follows from
Proposition
0\\
p((\
same
0
5.3.28 that
(\
0
'^'iU ojj U for all ^ G R. The
0
is true for
=
(^ J)
0
and hence cr^ leaves the
subalgebra
={(''ö' ^1)^ ^"^^} invariant. But p satisfies the KMS condition with respect to the defined by
^H 0
\(O^(AU]
0
^22;
Therefore, it follows from the uniqueness
<((
0
An 0
0
^^(^22)
0
V
\\
Statement of Theorem 5.3.10 that
(a^(An) l
A22)
automorphism
0
ar(^22)
0
Now, from the relations
oWo o\_ lojloij
/l
0
oWo
0
loojlio
it follows that 0
0\WO
1
00
0\
A .,
J==
0
0\ 0
p((Q
0
l
0
h'
group
128
States in Quantum Statistical Mechanics
and hence there exists termined
a
cr-weakly continuous one-parameter family t^^Tf
^
SUt de-
by
Applying rf
0
0
0
0
r,
0
1
0
to the relation
0
OWO
1
0
Jl
OV
l
/O
j ^l
0
l
OVVO 0
j
0
V l
0
it follows that
r,r;
=
i
=
r;T,
,
i.e., each F/ is
unitary. Similarly, from 0
0^/0
0
A
oW^
oWo
r
[l OJlo OJlO
~
O/'
it follows that
fff (^)
=
r,<(/i)r;'
.
Finally, 0
0 -
0
r,+.
0
0
0
0
1
0
r.
0
^p
0
0
r,
1
0
0
o\\
/
0
0
^
We conclude the
amination of
analysis of (T, ^)-KMS disjointness properties.
Q)J~
Vr.<(r.)
states for
finite
0
ß
with
an
ex-
In Theorem 2.7.17
we stated, and sketched a proof, that a von Neumann algebra 9K with a faithful normal state co is purely infinite, or type-III, if, and only if, cr^lgrjj^ is not an inner group of automorphisms of ^E for any nonzero projection E in the center 9}ln 9K^ In fact it is this latter characterization of type-III von Neumann algebras which is most useful in proving that a given von Neumann algebra is type-III, and for our purposes it could be taken äs a
definition. Theorem 5.3.34 shows that this definition is choice of faithful, normal state States
are
highly
co.
independent
of the
The next theorem shows that the
particular
(T, ß)-KMS
sensitive to variations of the temperature, i.e., variations of
Theorem 5.3.35.
^.
be a C*-dynamical System, and siippose that CO] corresponding to t\vo different valties ßj,ß2 ^ ^ Assiime that Uo^^ (^)" is a type-III von Neumann algebra. It follows that the states coj and (02 are disjoint. and
CO?
PROOF.
and
are
Let
Recall from the
only if,
(^, T)
KMS-states
Ttoj,
beginning
and TC^J,
have
no
of Section 4.2.2 that coi and co2
are disjoint if, quasi-equivalent subrepresentations. Suppose
129
KMS-States
ad absurdum that 71^^, and Tio,, have quasi-equivalent subrepresentations. Then it followsfrom Theorem 2.4.26 thatthereexistprojections^:/ G 3üj, T^a)i(^]" (^"^ T^CO^^^]' =
isomorphism (y.\na^^(^]"E\^^Tia3.>(^]"E2 such NO W ß\ ^Q because if ßi =0 then d)\ would be 0 then tradicting the type-III assumption. If ^^2 and
an
that
aL(na^^(A]E\)
=
7if^,(A)E2.
Tio;, (^)", conwould be properly
finite trace
a
on
Tiü,, (^)"Ei infinite and TC^JS (^)''^2 finite, a contradiction. Hence we may assume ^2 7^ 0- ^Y Corollary 5.3.4 it follows that T extends to groups f' on 7ia^.(^)" with respect to which =
the normal extensions co/
fj(^i) (5I)''^i such
follows that 9JJ
-
TCo,,
=
satisfy
the
(f', j5y)-KMS
an
that
Ot(Ti^^(A)Ei) for^
By Proposition 5.3.28 it automorphism group a on
conditions.
E\ for all t, and hence T^ defines
=
7rcü,(Tf(^))'i
e^.teu.
Next, Proposition 5.3.33 implies that the faithful normal
states (p^ defined
on
9JI
by
^l(^)
fA\
for i
=
^2(a(^))
fA\
^^(^^"'TrTpM C0l(^lj
^2(^)=-r-7TrYC02(^2J
'
,
1,2, satisfy the (er, jS^-)-KMS condition. This follows because
a,(A) for A G 9[R. Theorem 5.3.10 then
=
(a-' f?a)(^)
implies t
is the modular group for co/, i cocycle P in 5[Fl such that
=
^
ff-ß.t
=
(j\
1,2. Hence, by Theorem 5.3.34 there exists
o](A]
=
a
unitary
T,al(A)T:
for all ? G [R. As the state (p^ is both a^- and a^ -invariant, it follows that
C!>,(AY,}
=
=
=
for all ^ G 901,
i.e.. P/ is contained in the centralizer for (p^. Hence that 0-] (P^) P^ for all s. It follows that
Proposition
5.3.28
inner group of *-automorphisms of 307. Hence, by rescaling, a] is an inner *-automorphisms. It follows that 9[R is a semi-finite von Neumann algebra,
which
implies
=
p,+,-pX(r,)-r,p, i.e., ^H-^P/ is
a
unitary representation (T(^,_fc),
is
of R. But
=
now
r, ^r;
an
of is
=
,
a
group
contradiction.
Theorem 5.3.35
asymptotic
applies
in
particular
to KM S states
satisfy ing
abelianess condition of Theorem 5.3.32. Recall from the
that theorem that if
co
is
a
(T, jS)-KMS
.^i^
^l
S-T-^00 i3
exists for all ^,5, CG 91.
l
J5'
state where
j9 ^ 0, then
dto4AT,(B)C)
the weak
proof
of
130
States in
Corollary let
be
0}
a
Quantum Statistical Mechanics
5.3.36.
(^I,T)
Let
(T, ß]-KMS
lim /-^cx)
o
for all A^B^ C,Z) mth
G
be
^l. Let
C'-dynamical system,
a
^/ l
o
ß
e
K\{0},
and
dto,(A[,,(B), C]D)
=
0
(*)
J^
(^^, TT^J, l^co)
b^ the
cyclic representation
associated
CO.
It follows that there exists
(1) (2) In
na,('il]"E
n,o(^]"(^
particular
be States
,
is -
Ti(^(^)" n7ioj(Ul)^
E G
projection
a
on
E]
assume
is abelian.
that 5J does
not
^
=1,2. Itfollows that
PROOF.
CDi and CO2
Assume that
such that
type-IIL
have any characters and let coi and C02 at values ßi,ß2 ^ ^, where
satisfying the i-KMS condition ß^ ^ ß'j. Assume furthermore that co/ satisfies i
let
^ such that
state on
co
Center of 9Jl is contained in
T
(*) if ß^^O for
disjoint.
cire
satisfies the
the normal extensions of co and
condition
hypotheses
to 5[R
of the
corollary
and let ü and f be
Tr^X^)", which exist by Corollary
==
5.3.4. The
90r by Proposition 5.3.28, and it follows from the proof
of Theorem 5.3.32 that condition
is
(*)
m?
equivalent
to
mr^ 9JI'
=
where 30^ is the
fixed-point algebra of 501 under the action f. Now, let 11 " be the largest projection in 9JI n 9J1' such that T|SJJ{/| E] i^ ^^ inner group of automorphisms. Then Tifjj(^]" E is type-III by the remark before Theorem 5.3.35, since f is a multiple -
_
of the modular group. We may henceforth t ^-^ Ut'in "^ such that
E
assume
=
^. Then there exists
a
unitary
group
i,(A]
=
U,AU^^
.
But then
i,(U,] and
=
U,U,U_,
=
U,
so
U,
e
9jr
=
9J1
n
9Jl'
.
Hence
f,(v4)
=
U,AU'^
=A
for all A e m, But then m
=
m^
-
9JI
n
m' ,
i.e., y^l is abelian. In particular, if ^ has no characters then we must have " 11 in the above construction, and thus the last Statement of the corollary follows from Theorem =
5.3.35.
KMS-States
131
5.3.3. The Set of Ground States we analyzed various properties of the set Kß of C*-dynamical System (91, T). This analysis was restricted to the case j8 G IR and our next aim is to make an analogous analysis for 00. Since both these latter cases are similar we examine only the set K^Q ß of (T, oo)-KMS States, i.e., the set of ground states of (^, T). It follows directly from Definition 5.3.18 that K^Q is a weak*-closed convex subset of the state space E^. But KQQ need not be a simplex in contrast to the i for all t, and case of Kß with ß finite. An example is given by the choice T/ thus KQQ E^. Recall that E^ is a simplex if, and only if, 9l is abelian, Example 4.2.6. However, K^Q has one simple geometric property not generally shared by Kß, it is a face in E^ (see Remarks 2 and 3 after Theorem 5.3.32).
In the
previous
(T,j8)-KMS
subsection
States
over
a
=
=
Let (91, T) be a C*-dynamical System and assume that identity. It follows that the set K^Q of i ground states is convex and compact \veak^-topology and K^ is a face in Es^iLet CD e ^00, ^nd consider the follomng conditions:
Theorem 5.3.37.
9l has
an
The
(1) (2) (3) (4) (5)
is U-abelian.
pair (9l, co)
7Cft,(9l)'w
in the
abelian.
There exists
a
There exists
a
There exists
a
unique maximal unique maximal unique maximal
It follows that
(1)
If any of these
conditions
^
(2)
(3)
<^
fe
<^
ju^ in //^ in
MCO('^).
measure
^x ^^
Mü(^3i).
(4)
fulfilled,
are
=
/^
=
M(jc,(Kr^).
measure
measure
^
(5).
then
to
=
/^c
,
where ^^ is the central measure in M(j^(E^), and these measures are pseudosupportedby the intersection ofthe ground states, the R-ergodic states and the pure states.
In
particular, if co
is
an
extremal
7C()" PROOF.
A^oo is
2.3.15. If
O)
G
convex
and compact
K^ and
(p G
ground
=
by
E^ with (p
J5f ()
state, then
co
is pure and
.
the remarks before the theorem and Theorem <
^(D
then, by condition (4) of Proposition
5.3.19,
cp(,f(ArTf(A))
0 <
<
^co(^f(AY^f(A)}
for all same
=
fe ^ with supp /C (00, 0), and all ^ G 9l. Hence condition. It follows that KQQ is a face in E^.
Let
0}
e
KOQ be
presentation.
a
fixed state, and let
G
71^(91)''
by
the
be the associated
re-
cpisa ground
(Scy,7rcü,ßa), ^w)
Then
U^(t)
0 state
132
States in
for all
Quantum Statistical Mechanics
[R, by Proposition 5.3.19. It follows that
^ G
n,,(^)' and hence
(1)
Proposition,
r:^
and
=
n^(^)'r^U,,(U)'
(2) by Proposition 4.3.7. Furthermore (2) <^ (4) by the same (2) <^ (5) by Theorem 4.2.4. Since K^ is a face in 'STI, we have the
identification
M,,(K^) and
(3)
^
(5)
M^(E^i}
=
follows.
Now, if condition (2) is fulfilled,
7r,,(^2I)'n 7i,,(^l)"
we
=
have
K^(^l)'
=
n^(^)'n U,,(U}'
and hence ^c
But J.LX i^K Since this
by
=
=
/^x
=
^^E
the facial property of KOC is maximal in M(jj(E^^), the
measure
pseudo-support property
is
a
(3)
is
consequence of Theorem 4.1.11.
If
co
G
^(^oo)
valid. This
then /ij^ ö^o is unique maximal in Mco(K^). Thus condition condition (5) and the identification ß^ (5^0, i-e., w e =
implies
((E^i).
=
Note that the implication (2) =^ (1) in Theorem 5.3.37 is not true in general. simple counterexample is provided by taking T to be the trivial action on a non-abehan C*-algebra ^. Then any pure state on ^ is a ground state satisfying condition (2), but (^, co) is not IR-abelian if dim(^) > 2. However, the implication corresponding to (2) =^ (1) is true in a global version of the theorem. This global version can be formulated for more general faces F in E^^ than the face of ground states and also for more general groups G than R. Note in particular that if co G E^ and F is the closure of the set N^ of Tico-normal T-
A
invariant states
^, then
F satisfies the
hypotheses of the following theorem, partial generalization of Theorem 4.3.9. We is only assumed to be a face in E^^, not necessarily in E^.
on
and therefore this theorem is
emphasize that
F
a
Theorem 5.3.38.
Let (^, G, T) be a C*-dynamical System, \vhere G is a locally compact, (j-compact, topological groiip and ^ has an identity. Let F be a closedface of E^ mth the property that ifcoeF and \l/ is a unit vector in the space 'cücü of U^o- invariant vectors in )^, then the vector state d definedby
CD'(A)
=
(ilj,7i^(A)ilj]
,
^G^
,
is induded in F.
The following Statements
(1) (2) (3) (4)
The pair
{7i,,(^) F is
a
(^, co)
u
are
Ua,(G)y
is abelian
state (D in F is
inf A'eCo(TG(Ä]] all
for all CD e F. for all co G F.
Simplex.
Fach extremal
for
eqiiivalent.
is G-abelian
A, B e^.
vveakly cliistering
\o)(A'B)-(D(A)o}(B)\
=0
in the
sense
that
KMS-States
Any state CD in F such that {71^0 (^) u [/f^ (G)}' is a factor is ergodic. If o^\ cind 02 ^^'^ ergodic states in F and (D\ ^0^1, then the covariant representations (,^, 71^;, ^^0,) cind (0^.' ^^^2^ ^w.) ^''^ ^'^ot unitarily equivalent (äs covariant representations). If o:)\ and coi cire distinct ergodic states in F, then the face gener ated /)^02; by CD\ and co2 in E^^ is equal to the convex set {/coi + (l ^^ ^ [0^ 1]} generated by coi and 0)2,
(5) (6)
,
(7)
The
Remark.
,
special assumption
on
the closed face F used in this theorem is
necessary but the theorem is true without the
(see
133
Notes and
We will prove the
PROOF.
assumption
that G is ci-compact
Remarks). following implications
ii\
/ 6
=> 2 <^ 3 by Proposition 4.3.7 and Theorem 4.1.15. To prove 3 =^ l we technique closely related to the one used in the proof of Theorem 4.3.9. By
We have l
exploit
a
the latter it suffices to show that
A?,? is
Simplex,
a
where
A^^^
states
with their
=
onto
N^^
[ncoWE,,]
the
n^(^)"r^U,(G)'
G
(7,o-invariant
vectors in
g^^,
over
^,
,
and
UcoC^)"
continuous extensions to
a-weakly
2.4.21 that all co' e
{co';'e7^,'(Fj=l}
denotes the i-invariant TT^.; -normal states
FCO
E(a is the projection
=
.
we
identify Tifo-normal
It follows from Theorem
have the form
co'(A)
=
^(^,^n^(A)^,)
,
^'> l
where
^/,
G
,^
hence F^^c^^. states
and
^f,^^ \\^i,\\-
^^ for allÄ:.
1. But
=
But this
a)'(F,o)
means
=
that co'
l
l for all k and implies oj^^(F,^) be approximated in norm by =
can
co" of the form n
w"(A)
^(n,in,(A)n,u}
=
k= l
where and
Eco^k
we
^h ^^^ ^k ^
'^(ü(^)'
-
But then co,,^. G F
by
the
special assumption
have
o;'
<
^ lir.lpco,,, k= l
by
Theorem 2.3.19. Since F is
a
closed face, it follows that CD' G F. Hence
N^o^F
.
on
F,
134
States in
But since
A^^^
G-abelian
by
(2)
(5):
and
^
Quantum Statistical Mechanics
is
a
face of the
it follows that
simplex F,
the argument used in the
proof
A^^^
is
a
simplex,
and
(*i(, oj)
is
of Theorem 4.3.9.
If{7c,^(*sll)ut/,,(G)}'isanabelianfactor, then{7ü,,(^)ut/o.(<^)y
=
C1]
is
ergodic by Theorem 4.3.17. (5) => (6): Suppose that (6) does not hold, i.e., suppose that F contains two distinct extremal covariantly equivalent states coi and co?. Let co (coi + co2)/2. Further let T be the unique positive element in TifoC"^]' such that W
==
(D,(A]/2 for all ^ e 51 TG
Ua,(G]'
(Theorem 2.3.19).
=
It
(^,,,K,,(A]m,,] follows from the G-invariance of (ß\
that
and hence T ^
{K,,(^]^U,,(G]}'
.
Furthermore
co2(^)/2=:.(Q,7r,,(^)(l-r)0,,) and hence
A^(^,,,Ti,,(A](^-T]TÜ,,] is
a positive, G-invariant, linear functional dominated by both (D\ and co2 and it is, therefore, a multiple of both coi and co2 by extremality. Since (D\ ^ 032 it follows that (H T)T 0, i.e., r is a projection. Hence (^^, T^^, t/fo,^^^) is unitarily equivalent -
(,^, eS,02'^a), e7r,,t/o;,et/r^,,aa;, eQco2/\/2) and, therefore, to (,,, eS,^,, ^/V2) where ^ is a unit vector (distinct from t^oji 5^wi ^ojn ^fo, exp{/ö}Q,^, for all 0 G [R) in E^^,^. As {7r,(^)u^,o,(G)}' Cl, it follows that { 71^^(91) u L/c^ (^)}' ^2, where M2 is the algebra of 2 x 2 matrices. But M^ is a to
T^oji
=
factor, and
(7)
^
exist distinct
(coi
x co2)/2 is not ergodic. Hence condition (5) is not fulfilled. (6) does not hold, it follows from the preceding argument that there ergodic states coi and 0^2 in P such that
co
(6):
=
If
{7r,(^)u[/,,(G)y~M2 where
co
morphic
=
(coi
+
to the
co2)/2.
Hence the face
,
generated by
and co2 is
(ß\
affinely
iso-
positive
2x2 matrices of trace one, and hence isomorphic to the three-dimensional unit ball by Example 4.2.7. Thus condition (6) is not fulfilled.
(6)
=^
(7):
If coi and 0^2
are
distinct
ergodic states in F and property (6) holds, proving (5) => (6) that
then it follows from the argument used in
[{7i,,^e7i,,,(^); ^G^}u{C/,
e^,(^);ö^GG}]'-{air
+
a2(1]-r); (Xi,a2eC}
where T is the
projection from ^, © g^, onto ^^j As the face generated by co] affinely isomorphic to the subset of positive operators of trace .
C02 in
E^
assertion
(6)
^
is
and one,
(7) holds.
(4):
Assume that the state
clustering, i.e., E^,^ orthogonal to Q^j, special assumption
co
of condition
has dimension
larger than one. then the state coi corresponding on F and coi ^ co. By ergodicity
{7:(9I)u(y(G)}"
=
(4)
^(J
and hence
7to,(ai)"
=
ä'iE,.,?,,)
is
ergodic,
but not
weakly
If ?/ G "^00; is a unit vector to ^ is contained in F by the
.
,
KMS-States
this
But
(0),
^coi 5 condition 5
implies that ^ is cyclic for 7ia)(^), and hence the representation ^cü, ^wi ) is canonically unitarily äquivalent with (S^, TÜCO, ^w, ^w). Hence (6) is not fulfilled. ,
In Order to show
unpleasant
135
(4)
proof of the theorem we need an point of this lemma is to circumvent possible nonseparability of ^.
and finish the
(1)
=^
measure-theoretic lemma. The
difficulties which could arise from the Lemma 5.3.39.
Let
eMa}(E<$i) {An}^i a sequence a
li
be
co
probability
state
a
a C*-algebra barycenter co.rj a
over
with
measure
91
with
vector in
identity, ^, and
in 9l such that
^\\n^(An)^o,
-
YI
II <
-4-
00
.
n>\
(1) (2)
It follows that n(p(An}Q.(p li-almost all (p G E^. IfBe^ let
(Dl(B) Then
(p\-^o}^^(B)
is
If {A'^}^^^
(ri^,n^(B)ri^)
=
fi-integrable
a>l(B) (3)
converges to
=
vector
some
rj^^
^ for
G
.
and
J dt,((p)col(B)
.
is another sequence in 91 such that
5^||7r,,)Q,,-/7||
<+oo
,
n>\
then PROOF.
7Lfp(A'^j)Q.(p Let
(1)
fn((p)
=
be the
/
^^pfor iJi-almost
converges to
non-negative
\MAn -^4-1)^^1
-
all cp G "31
continuous function
(p((An -^+l)*(^
on
.
E
-^+l))^/'
.
Then
j dß(cp]fn(9? j dlJi(cp](p((An-An+lY(An -^+,)) =
-
Hence
^/
is
a
03((An -An+lY(An -^+l))
convergent series in
L?-(^]
^fnW
=
\\n^(An
-^4.i)Qa,|P
and therefore in L^
(/.i),
and
.
consequently
<+00
n>\
for
/i-almost
converges to
(2)
all cp ^ E<^, For any such limit ?/^.
(p,7r^(^)Q
some
Let
gn(cp]
=
\\n^(BAn)^^\\
.
is
a
Cauchy
sequence which
136
States in
the
By
Quantum Statistical Mechanics
triangle inequality
\gn(9)-gn+l(
Cauchy sequence in L^(i^i) converging {gn}n>\ Thus this function of (p is square integrable, and is
so
a
/*
jd^i((p)cDl(B-'^B)
lim
its
i-
(3)
e
(D(AlB"'BAn)
=
=
col(B*B}
.
oo
positive part,
<(S) for all B
co;j,(5*5)^'^".
f
;z
spanned linearly by
to
lim^Jd^L((p)gn((pf
=
=
Since ^ is
pointwise
it follows that
jdn(^]wl(B)
51.
The
same
argument
äs
in the
proof of (1)
shows that
||7I,,K-4)"'p)II^O for
/i-almost
all q) G E^n^ and hence
||7t^)a-^?J|^0 for
/^i-almost
END
all cp G ^^21.
co
-F, let ^,y4^
G
Since F is
THEOREM 5.3.38
It remains to prove that (4) ^ (1). ^ and let ß G M(a(F] be a maximal measure representing co. face in 's?j,/^ is pseudo-supported by the ergodic states in F, Theorem
OF PROOF OF
Let
a
G
4.1.11. Let ^ be
a
unit vector in
^üjw
^^^
{^n}
a
sequence in 31 such that
l|7i,,(5)a,o-^ll<2-''-^ By Proposition
4.3.4 there exist elements
S>,(U,.,)
=
Y,XlU,(g",] i= l
in
Co(Ua,(G))
such that
\\S,n(U,,]Ti,,(ABn)^,,-E,,n^(A]n\\
<
2"''
and
||5;;-(t/,o)7l,,(.l'5)Q,,-E,7l,X^O^||
<2-"
-
Novv, define A
=
5;.. (T(^5))
The relations above may be written
,
4
=
5;. (T:(A'B))
.
äs
\\n^(Si'(x(AB)))Ü-En(A),1\\
<
2'"
and
\\K,.,(S^;.(T(A'B}))n,,-E,,njA')ri\\<2-"
.
KMS-States
Lemma 5.3.39
By (p
there is
(1)
in
respectively,
Baire subset E of F with
a
and
eE,n(p(Bfj)ü^, 7i(p(An)O.(p
7r
l such that for
=
limits
ri^.^^p,
and
c'^
has
one
--l
\\7l^(lg(Bn))^o^-rj\\
\\U^(g)(7l^(Bn)^o^-^)\\<2
=
and it follows from Lemma 5.3.39 to
n(E}
some
^.
For fixed g E G
erywhere
converge to
137
that
(3)
7:^(1^ (5))Q(p
//-almost
converges
ev-
r]^. Since
n^(T:g(Bn))^g,
U^(g}n^(Bn)Ü^
=
U^(g]r]^
-^
for (^ G ^, it follows that
u
=
n^p
//-almost all f/) e F. Now, define ^
Then D is
a
=
{(Q. (p)eGxE', u^(g)n^
=
{(g^cp)
e G
X
; (p((T,(5)
Baire subset of G
Supports /z for each ^
G
^^} -5,)*(T,(5,0 -5))
=
F, and
x
G. The Haar
we
have
measure
on
follows from Fubini's theorem that ^Q <S) /^ is (p e F one has that
^^(ö')^^
=
/XQ
->
that
proved
0}
{cp
G
for
/z-almost Applying
all cp e F. similar arguments to
(^^
G
>}
D.
supported by
by assumption and it Hence for ^-almost all
n
jUQ-almost all ^ G G, and hence this relation holds for all ^ G G continuity of U^p. We conclude that e
F; (g, cp)
G is cr-finite
for
n
.
by
the strong
E^9)^
^'^
and
we
find
Baire subset EI C F such that
a
E\ Q E^{j. is supported by E\ and ^
^
particular, ri^
^^
^(P'0(p
if (/) G "1 is such that
=
(n^,^^)Q^
=
(Q^, ^^) Q^
F
is one-dimensional,
,
-
^lim^ (p(^)
n^
,
and
E^n^(A]Yi^ But
(p(An]
=
=
(Q^,7r^(^)f/^)Q^
=
^liTn^(p(^5)Q
.
(p (ABn) and thus
^^=E^n^(A)ri^
=
(p(A)(Q.^,r]^)Q.^
.
Similarly
r^
=
and hence
(>l^,n^(A')^^) for all (p e EI such that
F^^
=
is one-dimensional.
=
(ri,n^(A}Q
one
has
138
States in
Condition
ergodic
Quantum
(4)
state in
is
Statistical Mechanics
to the condition that
äquivalent
E^p
F, by Theorem 4.3.22. Therefore, if we
is one-dimensional for each
assume
(4),
it follows that the
set
{cp
G
E,-(n,p,Ti,,(A']^,p]
/
(^,^,71,^(4]^}
contains
no ergodic states. But this is a Baire subset of F, and it follows from the maximality of /.i that the set has measure zero (see Theorem 4.1.1 1). By Lemma 5.3.9 (2) and the polarization identity, it now follows that
(r],7ico(A')E,^7ia,(A}r])
lim
=
(naj(Bn)üa,, n,o(A'}nco(An)O.,)
n ^00
f di.t((p)(n,,(Bn}Ü,p,n,p(A^}n,p(A)ü,,)
lim
=
n-^oojf,
l d^i(cp](ri,^,n,(A']^,] /^A(((^)(^,^,7l,X^)C;)-('?,^co(^)^ro7ro.(^0^)
=
JF
=
-
JF
Since this is true for all rj e
E^o^^^,
it follows that
Ea,n,o(A')Eajn,^(A)Eo, for all A, A' e ^. The
pair (^,co)
Theorem 5.3.38 has several
of
groimd
states of
a
=
Ea,iL^(A]Ea,Tico(A']E,,,
is then G-abelian
interesting
by Proposition
corollaries. First
we
4.3.7.
consider the set
C*-dynamical System.
Corollary 5.3.40. Let (^, [R, T) be a C^-dynamical System where identity, and let KQQ be the set of ground states for the system. The follomng 15 Statements are equivalent: (1) (2) (3) (4)
77?^
pair (^, co)
7icü(^)' KQO
is
a
is
is abelian
U-abelianfor all for all co G K^Q.
Simplex. ground
Each pure
state is
inf A'^Co(rn(A]]
(5) (6)
G
an
K^Q,
weakly clustering
\CD(A'B]-(ß(A](D(B]\
in the
sense
that
-0
for all A, B e^. Any state co in KQQ such that n^jC^)' is a factor is pure. Ifo^i and C02 are pure states in KQQ then coi and C02 are either disjoint or equal, i.e., CD\
(7)
co
M has
Ifcoi
and C02
(D\ and
co?
in
are
E^^
by
co\
C02
or
(D\
=
C02
distinct pure states in K^Q then is
{Icoi gener ated
i
equal +
to
the
(1-1)092;
and coi-
convex set
AG
[0,1]}
theface
gener ated
by
139
KMS-States
in
obtainedby replacmg "pure" by "ergodic"
The Statements
(4'), (5'), (6'), (7'):
(4), (5), (6), (7), respectively, obtainedby replacing "pure" by (4), (5), (6), (7), respectively, 77?^ Statements
(4''), (5''), (6''), (7''): tremal in KQQ"
in
The set K^ is
PROOF.
It follows that if
co
"co is extremal in
theorem
is
Ä^oo"
are
to the
primed
We next show that the face F
5.3.38. If
is
co
a
Theorem 5.3.37, and hence AToo is
by
ground
state and
ij/
G
is
^ojöw
unit vector
a
(ll^,7l,,(A)ll^}
=
corresponding
(P^,P7ü,,P,P^,,P,l//)
{7i,^(^)^U,o(^)y
is the
projection
applies
to F
Proposition
5.3.19
(5)
^oo, and it remains
equivalent ground states. This is 5.3.19 implies that orem are
a
state
through
to
co' is unitary
,
defined
p-KWAl
defining
,
cyclic representation (S^y^^w, ^w',^') equivalent to the representation
It follows from
E^.
ones.
then the
G
face in
K.^ has the special property required in Theorem
=
CD\A)
where P
a
state, the Statements "co is pure," "co is ergodic," and all equivalent, and hence the unprimed Statements in the
ground
equivalent
are
face in E^i
a
a
"ex-
by
.
that co' is
a
ground
state. Thus Theorem 5.3.38
(l)-(7) of that the special case of (7), and since Proposition
to show that the Statements
of this theorem in the
to Statements of
(1)~(7) (1), (3), (4),
clear for Statements
U,oW
C
and
7l,,(^)"
ÄToo, the equivalence for Statements (2) and (5) follows immediately. (6) of Theorem 5.3.38 takes the form (6'"). If (jo\ and C02 are pure ground states and coi 7^ co2, then the representations (ÖWP^TCÜ, ) ^nd ((0^5^10,2) ^'*^ ^^ot unitarily equivalent. But since two irreducible representations are either unitary equivalent or disjoint, this is equivalent to Statement (6) of the corollary.
for all
G
co
Statement
The was
implication (1) ^ (6) in Corollary 5.3.40 is related to a importance in the development of quantum
of fundamental
result which field
theory,
theorem. This theorem demonstrated that the examination of the
Haag's ground states of a given dynamics necessitated the examination of a variety of unitary inequivalent representations. A more local Version of this implication is äs
follows.
Corollary 5.3.41. Let (^, !R, T) be a C*-dynamical System, and let be extremal ground states on ^ such that (^, co/) is U-abelianfor It follows that
CD l
and
co? are
COi
PROOF.
Assume that coj and co2
subrepresentations,
and since
i
either or
C02
are
not
Tif^.(^]"
disjoint
=
COi
disjoint.
or
=
C02
oji and co?
i
1.2.
equal, i.e., .
Then n^o^ and TI^), have equivalent Theorem 5.3.37, it follows that
J^(^^.) by
140
States in
coi and co2
Quantum Statistical Mechanics
quasi-equivalent (and even unitary äquivalent) Hence (coi + co2)/2 is by Proposition 2.4.27. Since the set of ground states is convex and the
are
.
a
factor state
set of
invariant states (p such that (^, (p) is [R-abelian is convex äs it follows that (coi + co2)/2 is a ground state and
Corollary 4.3.10,
is G-abelian. Theorem 5.3.37
factor,
one
deduces that
implies that 7i(cfj,+tü2)/2(^)' is abelian, (a)\ + co2)/2 is a pure state. Hence COi
The last
corollary
=
(J02
(<^l
=
+
<^2)/2
a
consequence of
(^, (CL>I
co2)/2) a
.
the set of all G-invariant states. It is
concerns
+
but since this is
a
partial
generalization of Corollary 4.3.11.
Corollary 5.3.42. Let (^, G, T) be a C*-dynamicaI System, \vhere G is a locally compact, a-compact, topological group and ^ has an identity, and define F E^ äs the set of all G-invariant states on ^. It follows that all the Statements (l)-(7) in Theorem 5.3.38 are equivalent for F. PROOF.
F
=
E^^
clearly satisfies
the
requirements
of Theorem 5.3.38.
We know that all extremal
ground states are pure, by Theorem 5.3.37. It is general ergodic state on a C*-dynamical System spectrum unequal to the whole real line, then co is either
remarkable that if
is
co
(51, [R, T)
a
with energy be decomposed into pure states with lower symmetry in analogous to that given in Theorem 4.3.37.
pure
or can
Theorem 5.3.43.
(^, R, T),
and
Let
presentation Uaj of U It follows that
co
be
CD
assume
is not
^-ergodic
an
that
state
on
a
the
spectrum of the the whole real line.
is either pure,
or
there exists
a
a manner
C*-dynamical System
associated
pure state
co
unitary
and
a
re-
positive
real number T such that
(A}=^ JQTdtcbMA}}
(1)
l
for (2) (3)
0} co
all
Ae^, is periodic with period T, i.e., and co o T/ are disjoint, co i co
Furthermore,
a
pure state
transformations by PROOF.
Let 931
=
co
co o
o
TJ^
T^,
with the proper ties
if
n,o(^liy'.
Since
u,(t)mu,(tY=m for all
/
R,
one must
have
u,,(t)m'u,,(tY for all
^ e
[R. Hence
one can
=
define
T on
=
9JI' by
m'
co,
whenever
0 <
(l)-(3)
t < T
is
.
iinique
up
to
KMS-States
Tt(A) But since
Uo,(t)AUa,(tY
=
Qf^ is separating for 901',
.
has
one
a,(m')
Aem'
,
141
a(Ua,l^,^)
=
in the notation of Definition 3.2.37. Now it follows from the
ergodicity
of co, and
Theorem 4.3.17, that
m'r^U^(U)' i.e.,
T
is
ergodic
50l' in the usual
on
implies
of Theorem 4.3.33
sense.
U^
=
(j^(^') ar(^')
that
is
Lemma 3.2.42. Hence the spectrum
whole real line there
a^C^')
Case L In this
case
=
are
only
,
Therefore, the argument used in the proof is additive. But cr^(9Jl') is Symmetrie by a
group and since
a(Ufo)
is not the
possibilities.
two
0.
the action
T on
501' is trivial, i.e., m'
U^(R)'
C
and hence
^'
'm'nu^(u}'
=
=
Ua^
.
It follows that Tioj is irreducible and co is pure. Case 2. ff.Cm') 2nI./T for some T > 0. =
It follows from
ergodicity,
in the
äs
of Theorem 4.3.31, that the
proof
spectral
subspaces
(yR'Y({2nn/T}) are
one-dimensional, and there exists K e
unitary operator
a
(9[R')^({-27ü/r})
such that the linear span of the powers K",
G
w
Z, of
K is dense in ^' In .
particular
501' is abelian. Note that
i,(V) and hence the spectrum of K is
equal
to the unit
a(F) The Gelfand
with
C(T), 2nt/T. Define
a
e'^^^^^V
=
-
T
circle,
.
transform, Theorem 2.1.1 IB, identifies the C*-algebra ( generated by V through the angle
and the restriction of T/ to ( identifies with rotation
probability
measure
//
on
^(c)
-
T
by
(QO;, CQ^)
for C e CC, where C denotes the Gelfand transform of C. normalized Haar
measure
on
T. Put F^,
=
[SJl'Qo;]-
By t-invariance,
If ^4 e ^ is
bounded i-spectrum, then
F^n,,(A)F^ is
an
element in
'^'F^^
G
F,,mF^
=
m'F,,
with bounded i-spectrum and hence
F,,7i,,(A)F,,
G
(:F,,
.
an
/i is the
element with
142
States in
It follows
by
Quantum Statistical Mechanics
norm
that
continuity
F,7ü,,(3I)F,, Thus,
if ^ G
^, there exists
a
unique
a
unique
=
yeJ^cb,(A) C(T) corresponding
cby
is
to A. Since
State for all y G T.
Now if D G G:
2.3.11.)
CF(,
function
in
a
.
C 6 ( such that
F,onj(A)Faj and hence
^F^
C
(If H ^ ^l,
ä)y(i)
=
=
C(y)
l and
d)y(A)
>
0 if ^
>
0, it follows that
approximate identity
one uses an
and
Proposition
have
we
(ü,,,n,,(A)DÜ,)
(^,,,F,,n,,(A)F^DQ,,) (Ü,,,CDQ,,)
=
=
l d^i(y]C(y]b(y]= Jjj d^i(y)ü,(A]b(y]
=
.
Jj Since ( is dense in 9}!'
9J1"
=
n
9JJ^
it follows from this
expression, Proposition
4.1.22, and Theorem 4.2.4, that
w(A) identifies with the central
decomposition Since
co
in this
äs
decomposition
of co, and this is
just
the
unique
extremal
case.
is i-invariant
/ di.i(y)d),(A)D(y) and
J^d^i(y)co,(A)
=
we
=
find
(^^,n,,(A)DQ,,)=
/
Jj
this is true for all D G
it
C(T)
27rA
di.i(y)(I).(i,(A))D(y
,
T
\
follows, by continuity, that
^y + 2Kt/T(A) =a)y(lt(A)) for all
^ G
[R,
y G
T, and
A e'^. Thus
we
co
and then the central
=
^
Since
co o a^ is pure for 0 < / < r, then co
/i-almost o
Remark.
all t,
dtäMA))
co
is pure. If
equivalent
decomposition
Statement follows from the
Since the
takes the form
.
JQ
a^, would be
contradicts the fact that the
uniqueness
coo
(= extremal) decomposition
CD(A]=l-[ some
may define
of
co
to
co were
co
o
equivalent
a^^.^ for all
is the central
uniqueness
s
to e
co o
a^
for
U, and this
decomposition. The decomposition.
of the central
decomposition in Theorem 5.3.43 is the central de composition corresponding measure is an orthogonal measure. Moreover one concludes, a posteriori, that the decomposition coincides with that of Theorem 4.3.37 with H TZ, but the //-ergodic states {ij^co; t G H} are in fact pure. Finally one can establish that the assiimptions of Theorem 5.3.43 do not imply that the pair (^, co) is //-abelian and hence the result is not a direct the
=
conseqiience of Theorem 4.3.37.
143
KMS-States
The
important
feature in Theorem 5.3.43 is the
uniqueness which
T
is
Statement for the
occurrence
allows
structure. The construction used in the
proof decomposition of ergodic
one
of
a
to conclude states of
periodic
strong Systems for a
periodic.
Corollary 5.3.44. Lei (21, IR, T) be a C* -dynamical System where with period T, and assume that CD is an R-ergodic state on 91. It follo\vs that there exists a pure state cb such that
co(Ä)=^ JQ[
dtcö(T,(A)}
translates cö
T^ and
T
is periodic
^
for all A G 91, and any disjoint. Furthermore, PROOF.
is
The
periodic
two o)
is
unique
(D
the
is pure
period
cboi^ ofco
periodicity assumption on period T, and hence
T
implies
by
that the
are
either
equal or
T.
unitary representation U(a
with
(7(Ua,) If
o
up to translation
we
T of
may set cö T
is
an
=
co, if not
C
2nZ/T
we use
integer multiple
of the
.
the cä of Theorem 5.3.43 and note that
period
T
occurring
in Theorem 5.3.43.
This last result partially complements the discussion of the uniqueness of the ergodic decomposition given in Chapter 4. It should, however, be emphasized that the periodic assumption which is crucial for the uniqueness of the extremal decomposition in Corollary 5.3.44 is largely incompatible with the conditions of asymptotic abelianness which are relevant for uniqueness of ergodic decompositions. More precisely, if (91, G, T) is a C*-dynamical System based on a compact group G one can establish that (91, co) is G-abelian for all co e E^if and only if the fixed-point algebra 91^ is abehan. The latter Situation is atypical if 91 is simple and G is compact.
5.4.
StabiMty
5.4.1.
and
Stability
Equilibrium
of KMS States
In the
previous section we analyzed properties of individual KMS states and properties of the set of KMS states. But hitherto we made no attempts to justify the Interpretation of KMS states äs equilibrium states of quantumaffine
mechanical Systems. The purpose of this section is to remedy this omission. We have noted several times that the finite-volume Gibbs states are the
unique
KMS states and that the KMS property,
modular structure, has
a
empirical acceptance of is
a
tendency
to
persist
or
at
least the
accompanying
in the infinite-volume Umit. Thus
the Gibbs formalism indicates that the KMS property equihbrium and this motivated the foregoing
suitable characterization of
analysis. Now, however, we attempt to derive the KMS condition from various general, physically motivated, properties, notably properties of ergodicity and stabiUty. As a prehminary we analyze the stabiHty properties of KMS states and ground states under perturbation of the dynamics. In Section 3.1.4 we discussed perturbations of generators of CQ- and Q-semigroups and we now apply this theory to groups of *-automorphisms T of C*- and ^*-algebras. We are particularly interested in perturbations by bounded symmetric derivations, and the algebraic structure leads to several refinements of the general theory. For example, the derivation property ensures that the perturbed group T^ is a group of *-automorphisms and hence it is norm-preserving by Corollary 2.3.4. This strengthens one of the conclusions of Theorem 3.1.33. Next recall that every bounded Symmetrie derivation (5 of a ^*-algebra 9Jl is inner by Corollary 3.2.47, i.e., there is an // //* G 9JI such that =
ö(A)
=
i[H,A]
for all ^ G 9J?. A similar result is true for representations of C*-algebras, Corollary 3.2.48. Thus, it is natural to concentrate on inner derivations. The perturbation series for T^ then assumes a distinctive form in terms of commutators and T, and
T^,
are
related in
a
simple algebraic
fashion
by
a
unitary
co-cycle.
Proposition denote the
define
5.4.1.
Let
infinitesimal
(^,T)
be
generator the bounded derivation dp by
a
C*-
of
or
W*-dynamical System
and let 3
P' e ^ Fiirthermore, for each P ^ and dp (A) D(dp) i[P^A]for A G 51. T.
=
=
Stability that d -{- dp gener ates
Itfollows
and
145
Equilibrium
one-parameter group of" -automorphisms
a
1^ of ^ given by
T^(A)^x,(A]+Y,f JQf'dti JQr dt2--- JQr"'rf?K(p),[--.K(p),T,(^)]]] ~^i
Moreover,
one
has
Tf(^) where
Pf
G
rf
9l is
=
rfT,(^)rf*
one-parameter family
a
l +
=
,
of unitary elements,
^ f Jo/' dt, Jor dt2--- Jo/'""' dt^,(p}
-
T,
determined
by
(P)
~^
=
1 +
Jo
t^ which
the
satisfies
/' dti r dt2---
Vf
Jo
co-cycle
(/')--< (P]
,
relation
YP ^
All
/'"'' ^?<
JQ
r^r (r^} t^t(^ s) ^
t+s
'
Integrals converge in the strong topology for the C"" -System and in the atopology for the W*-system. The Integrals define norm-convergent series
weak
of bounded operators
||Tf(^)-T,(^)ll
and
<
(ei'iii^"
lirf -i||
i)IMII,
-
(ei'iii^ii
<
-
1)
.
the
proof for the C* -System. The ^*-case follows by transposition proof of Theorem 3.1.33. The first Statement of the proposition can be obtained from Theorem 3.1.33 by suitable replacement of J^ by ^, etc. but it is most easily deduced from the proof of the latter theorem. This proof established that T^ is the unique solution of the integral equation. PROOF.
from the
We
give predual
äs
in the
/'c/r,[T(P),TTf_J^)]
^(A}=T,(A]+i
.
JO
Alternatively T^
is the
solution of the
unique
^,(A)=,,(A]
+ i
integral equation
l" dt,r^^T.([r(P),,,(A)])
.
JQ
Solution
by iteration gives
the
perturbation expansion of T^.
The
automorphism
property of i^ follows with the aid of the derivation condition, e.g.,
^t^,(<(^)^r(5))=o. Next consider
defined and has
parameter family t G
[-^0,^]
for
Pf
defined
norm
by
the first series. The nih term in this series is well-
less than
\t
"
||P||"/!.
of elements of ^ with
some
^ > 0. Next
.IpP
one
^=,Tr.(P),
FQ
=
Thus 1
.
Pf
is
a
norm-continuous
Consequently
has
.JT^P*
^=-.-.(p)rr,
Pf
one-
is invertible for all
146
States in
in the strong
Quantum Statistical Mechanics and hence
sense
^(rfrp^Q dt
FfPf*
Therefore,
follow from the
D and
=
Ff
is
unitary
co-cycle relation.
for
^^=iT^,
"'
äs
and
rf,J,^o
rf
=
.
(Ff)
But T,
^FfT,(F;) FfT;(Ff)|^,^Q
Moreover,
T,,
Ff.
'
'
we
/FfT,(F;)T._,,(F)
=
^-i^Ff,^^
Thus
Ff +
-
/
/
t
will
(P)
and
.
5'^FfT;(Ff)
first-order differential equation and boundary condition, for each the two functions are equal and can be obtained by Iteration of the
X,(s}
for other
first note that
and hence
/FfF;'(^)T,(T.(P))
=
=
F^^'^^^
-
[-^o,^]- Unitarity
t e
To establish this relation
satisfy
the
same
(R.
Therefore, integral equation
t G
ds'X,(s')i, + ,^(P)
Jo Next the
unitarity
and
continuity
norm
of
F^
ensure
that
/
^
Ff T;(^)Ff*
defines
a
continuous one-parameter group of *-automorphisms of 51. Let ö denote the infinitesimal generator of this group and SQi A for / 6 [R\{0}. (}i i -}- Ö -^ One has
strongly
öp^^B)
=
fT
(;.z
But
(Az
+
dY^
is
+
<5)(^)=(;.z + ^)(rfT,(^)rf)Uo 7tj -(FfT,(az + <5 + ^p)(^))Ff*)|,^o-^
bounded, by Proposition 3.1.6, and hence A
=
(}.i^
~dr\B]
=
(Ai^d
+
(5p)"' (B]
for all B 6 "il. It then follows from Theorem 3.1.10 that
Tf(^)//
Yim(i--(d + dp]Y\B) ->
//
which
gives
Now if
/
77
(l --5]"' (B]
lim
=
v
oc
00
v
n
=
J
the desired relation between T^ and one
morphisms is Therefore,
T
perturbs r^by
rfT,(5)rf*
T.
perturbation -P, then the final because it has the generator (b ^ bp] b. bp
one
a
group of auto-
-
niust have
T,(/!)
=
ff^f (/()ff*
,
where
rr-^+E(-^')" JO/'^^' ~^\
r^^2--- /'""^^.Tf(p)---<(p)
./o
JQ
Consequently
rf But then
=
(ff)*T,(^)(ff)
.
.
Stabil! ty and
^(ff)'=/Tr(p)(ff)*=/(ff)'t,(p) (f^)*
Thus,
and F^
condition at
^
Finally the
=
0.
the
satisfy
estimates
T^
on
-
=
(rf )*
equation
and
boundary
which is the second Identification of F^.
F^
and
T
147
.
first-order dififerential
same
Pf
Therefore,
Equilibrium
H
-
are
straightforward.
Throughout the sequel we use the notation introduced in Proposition 5.4.1 for the perturbed automorphism group T^ and the related co-cycle F^. This proT^ G Aut ^ is continuous with reposition establishes that the map P G ^ spect to the norm topology on ^ and the strong topology on the automorphism groups. If (^, T) acts on a Hubert space and T is unitarily implemented, one easily attains continuity properties for strong convergence. Moreover, the group T^ is unitarily implemented and the co-cycle F^ has a simple identification. i>
5.4.2.
Corollary
that ^ acts
Adopt
the
assumptions of Proposition 5.4.1 but also
Hubert space
on a
i,(A] vvhere Ut
Qxp{itH}
=
is
strongly
a
assume
and =
U,AU;
,
continuous one-parameter group
ofunitary
operators. It follows that
Tf(^)
L/r^c/f*
=
rf
,
=
[/fc/_,
,
where
Uf If, finally, P is strongly to zero,
-
r,)tA||
lim
0,
-
>CX)
for
fl//
PROOF.
i/^
If U^ is defined
äs
uniformly for
=
'^.
t
-
x,(A)m =0,
infinite intervals o/
in the Statement of the
^ and XQ
||(Tf"(^)
M \^^hich converges
->00
and A G ^,
G
.
of selfadjoint elements of
a sequence then
||(rf"
lim
QXp{it(H -{- P}}
=
=
iU^PU.,
=
corollary
[R.
and Xf
=
UfU-t,
then
iX,r,(P)
Thus, Xt is the unique solution of the integral equation Xt
=
[
i^i
dsX,i,(P}
.
JQ can be obtained by iteration and one finds Xt r^, where F^ is defined Proposition 5.4.1. The identification of T^ follows immediately from this proposition. Moreover
This solution äs
=
in
ll(rf'-i)i/'||<
l'
Jo
ds\\pu.,n
148
States in
Quantum Statistical Mechanics
But strongly convergent sequences are uniformly bounded and hence F^" converges strongly to the identity, with the appropriate uniformity, by the Lebesgue-dominated The convergence of T^"
convergence theorem.
to
T
then follows from the
re-
the Hubert space
T a
presentation
Tf"(^)^rf"T,(^)rf"* EXAMPLE 5.4.3.
Let
be the CAR
^l(l))
.
algebra
over
and
l)
one-parameter group of Bogoliubov transformations such that T^ ((/)) a(Utf) with Ut Qxp{itH}. The simplest Symmetrie perturbations are given by quadratic =
elements, a*(f)a(f]
a*(f)a(g)
+
or
combinations of such
a\g)a(f)
=
The group T^ resulting from transformations because for
a
(*(/
g)a(f
+
elements,
+
g)
-
e.g.,
a'(f
quadratic perturbation
-
g)a(f
is also
a
-
g))/2
group of
.
Bogoliubov
P=^Aja^(fj}a(f,) y=i
has
one
5p(a(f))^iy^lj(f,/))a(fj) /
.
^
7=1
It follows that
projection
on
tf (fl(/)) /), i.e.,
=
a(Uff)
with
Ejg for all
t/ G
=
t/f
=
Qxp{it(H
+
I.]^^ljEj)}
where
Ej
is the
(fj^g)fj
t).
Quadratic perturbations of Bogoliubov transformations of the CCR algebra can also be introduced for regulär representations but äs the creation and annihilation operators are unbounded these perturbations are only affiliated with the representation. Next
System
we
examine
(^, T)
KMS state for the
(T, ß}-KMS
of KMS states
stability properties
and establish that each such state has
perturbed
(T^, ^)-KMS
States and the
dence. If the KMS states
evolution T^.
co
over
vector state
a
Subsequently,
we
a
dynamical
co^ which is
a
deduce that the
states are in one-to-one correspon-
identified
äs equilibrium states of a physical stability of the pure phases, and the Overall phase structure, under perturbations. These results are obtained with various perturbation series which are easily understood in the finite-dimenare
System, these properties correspond
to
sional Situation. Let Mfi denote the space
,j
and for //
C*-algebra ofnxn =
//* G
matrices
i,(A)^e^^^Ae-^^^ The
perturbed
group
on
the /7-dimensional Hubert
M,j define the automorphism group
T^ corresponding
Tf(^)=ß'^(^
+
io P
=
T
by
.
P* G
^)^^-^(^ + ^)
M,, is then given by .
Stability
Moreover, by Example 5.3.31, the
and
Equilibrium
149
state
Tr>2^
"(^^=Tr,M^) is the
unique (T, jS)-KMS
state and the state
co^(Ä} unique (T^, j5)-KMS
is the
i:T^^^(e-(^(^ + ^^A) =
Tr(^"^^^^^^)
state. But
co^
be
can
in terms of
expressed
in
CD
several ways.
First,
,,^,_co(Ar^
ca''(A)
=
^c^-ß} where
l^P ^^-ß(H + P]^ßH i.e., the
r^ is the analytic extension of the co-cyde F^, which relates i^ and i, at point iß. This identification follows simply from cyclicity of the trace. But
one
has
Ff,
-
l +
V(- l )" ^l
/
ds,
JO
r dS2
'
/'""' dSn
T,,
(P)
.
T,,,
(P)
JQ
JQ
and this allows co^ to be
expressed äs the ratio of two perturbation series. This rearranged äs an expansion of co^ in terms of co. Somewhat surprisingly, the terms of this latter expansion are integrals of truncated
ratio
can
then be
functions.
Similarly
one
finds the
Symmetrie form
for CD^ in terms of
CD
.,.,_co((rf,,,)*^(rf,,,)) ^^ -((r^,/,)*(rf,/,)) "
and this yields a different set of expansions. This Symmetrie expression is particularly useful for the study of the associated representation because one may choose Q^^p such that
a/
'^co(^iß/2}^oj -
l^f^(r^/2)^w| and then deduce
of the
expansion cyclic vector. Stability properties of general i-KMS states can now be analyzed by ploitation of these algorithms for a;^,Qf,/, etc., and the perturbation series the
Ff
series
an
to construct
are
i^-KMS
states. In the
evidently well-defined
and
ünite-dimensional
uniformly convergent
this is less evident. The necessary estimations KMS condition.
case
are
case
ex-
for
the various
but in the
general
consequences of the
r-
150
States in
Quantum Statistical Mechanics
The
major features of stability of KMS states are shared by C*- and ^*dynamical Systems. Discussion of the J^*-case is however more complex because it depends upon approximations in the strong operator topology while the C*-case can be handled by uniform approximation. We next state the common results, give the detailed proof for C*-systems, and outline the proof for )^* -Systems. Theorem 5.4.4.
Let
Hubert space
be
(^, T)
such
C*-
a
T,(A) vvhere Ut and fl State,
a
=
exp{/Y//}
is
on a
=
U,AU:
,
strongly continiioiis one-parameter iinitary groiip, cyclic vector such that the associated vector
a
normalized U-invariant
(jo(A) (Q, ^Q), is a following Statements
i-KMS
=
The
(1)
W-dynamical System acting
or a
t hat
//P
=
^(z)
P* G =
for all
(P)
T,
z
=
X)[,"^
^, then Q
(zi,
.
{z;
=
T,j
-
.
a
D(^(z)),
G
(P)
< Im zi <
The vector-valued function continuous and
strongly
.
.
.
p^-L-i//
^_i /9 defined by
Im Z2
< Im Zn <
,^(z)Q.
i|pir
<
0}
holomorphic
is
boiinded
uniformly
ii.^(z)fi|i
sup
where
e^^"Hp^(zn-^-z,^}Hp
=
in the tube
,z,j)
.
state.
valid:
are
on
.
in the tube X
its closure
T)_jp,
|/^, and
.
--^^':;/. (2)
IfP
=
p* 6
has the
Q^
-
21, then Q
strongly
1} +
convergent
is
a
i^-KMS
^^1
state
(3)
state
is
is the
Q''
=
dSn T/. (P)
...
.
T,,,
(P)0
a
.
=
bounded
and it is the
unique i^-KMS normal
state
if,
and
factor.
.
.
,
tn]
=
=
holomorphic on
.
(ü'-, A^P) (Q^ QO
P* G
^, the truncated function
CDT(A, it,,(P],
.
.
.
,
T,,
(P))
boundary value of a function FA(Z)(= cor(y4, TZ(^))
vvhich is
e^^ + ^^^l-Q.
(jf defined by
For each ^ G Ul and P
F^(^i,
vector
J-1/2 <5i<-<5<0
(o''(A)
only if, ^r^
and the
perturbation expansion
/
V ,,>!
Moreover, the
D(e^"+^^^'-)
6
its closure
in
the tube tub(
X'^j,
and ,
D_p
continuous and
^^^i
(P)))
uniformly
Stabil! ty and
\F^(z]\ <2^n\\\P\nA\\
sup
Equilibrium
151
.
zeD^^'J <
l the
uniformly convergent
series
Moreover,
//2||P||
of(A) =co(^)
+
perturbed
(jf is determined by the
state
E/ ^-i<^i<-
^<5/,<0
>i
xdsi'" dsn COT(A, %, (P),
.
.
.
,
T/^i
(P))
and hence
lim||co^'
-
co||
=
0
a
each net PO,
for
Remark.
=
P^
The convention
e
^ such that
\\Py\\
>
0.
l which we adopt in the theorem is conß later applications. With this convention the for slightly confusing the addition of the "energy" P to the Hamiltonian to perturbation corresponds the of the modular //, i.e., generator unitary group. If, however, one rescales by with and replacing T/ simultaneously replaces P by ßP, then H is replaced i-ßt H P. the Thus, by perturbation corresponds to the subtraction of the "enThis is P. most ergy" easily seen by examining the perturbation of Q. With l has one ß =
venient but
Q^-^(^+^)/2Q But the
.
to the
rescaling T^ i-ßt corresponds general ß one has \-^
replacement
ßH
H\-^
and
hence for
^ßP
PROOF.
^
^(-ßH + ßP}/2^
^
^-ß(H-P}/2^
^
We examine first the C* -System.
First choose PL
(1)
possible by Proposition
e
^^ such that ||Pi||
2.5.22 and
||(^,(Z)-^M(Z))Q||
<
one
<
||P||
and
\\PL
-
P\\
->
0. This is
has
^||T.(P.)...T,XPl-PA/)---T..(PM)ß||
-
y=i
For
three line
the right-hand side can be estimated by repeated application of the theorem, Proposition 5.3.5, to the vector-valued functions
where the
ß/
z
e
^_i/9
F(z)-T,(a)---Tz,(ßi)n value of the
e
^T
norm
Si =S2
where
si
=
,
subsequently identified with P/,, PM, etc. Thus, the maximum of F is attained at one of the + l points
are
=
'"
=Sj
Im z/. But at such
=
a
--,
point
Sj+i =Sj + 2
=
=Sn=()
,
152
States in
Quantum Statistical Mechanics
|F(/i -zy2,...,ry-zy2, /y+l,...,/)|<
liail'
=
liail'---llö7+ill'l|T,(ßt)---T.xß;)^ll'
llöy+i ll'l|T-./2(Tr,(a-)
^Jßi))^ll'
7-1
The
equality of the second line Symmetrie form derived
in the
follows from the t-KMS condition (at value in
Proposition
iiG^,(z)-,^M(z))oii
5.3.3.
Therefore,
one
ß
=
-l)
has
\\pL-pM\\i^\\pL\r' \\pM\r
<
7=1 e X)^"^,p. Henee ^L(Z)Q. is a Cauehy sequenee uniformly on D^'l^'j/^. Beeause uniformity and the ehoiee of PL G ^T the limit function Q(z) has the holomorphy and eontinuity properties stated for -^(z)O. Moreover, by the same estimates
for all
z
of the
||D(z)||
lim
<
\\^L(zm
\\P\\"
<
L^oo
for all
G
z
X)^"\
It remains to
identify Q(z) and .^(z)Q. Corollary 5.3.4 and Theorem 5.3.10 that A the pair (^'',n) and in partieular 5IQ
/^.
It follows from
for
operator
-|
< Im
z
=
c
e^ is the modular
D(A'")
whenever
0. But A'" is closed and hence
<
A'"^PQ
A^^PL
lim
=
O
.
L^co
This ?^
establishes
-
l.ln
ü(z}
p^^i(zN-,-ZN]p^ But
one
^(z)Q.
=
for
partieular -
.
.
z
for
n=l.
A'^-^'^^'-^P^D
assume
the
identification
for
-^
PA'(-''^-'~^^^P- PA'^-"'"-^^^PQ .
also has
^/z;vp^^/(-^^-i--v)p^ by the previous estimates. Thus, The bound
on
||^(z)Q||
Assume P
(2)
Next
'^^^\/j
G
G
A'("'-"^)p^Q
.
A'^'^ is closed, by ehoosing Qi
^^ and define
,7^
.
äs
follows
r^ '^+y](-i)"
.
one
=
Tj^.
^
O(z)
obtains
P in the
Q(z)
previous
=
,^(z)Q.
set
of estimates.
by
r^^i r ds2"' Jor"'j^T,,,(p).-.T,,,(p)
Jo
.
Jo
Each term in the series is well-defined because P
formly,
and
eludes that
a^ Of
T^,
G
^
^^.^2^-
Now
caleulating
G 31^, the series eonverges uniwith convergent power series one eon-
^-r and
T,,(rf;^)
=
Tf,(^)r^
for all ^ G ^IT- Therefore,
(r!,/,^, Tf/,(^)T!,/2(s)r!,/2fl)
=
=
(T-,y2(r5/,^*)a, T_,y2(r!*/,5)n) (r!,y2Q, 5^r!,y2n)
for all y4, ^ G "^IT, where the second Step uses the Symmetrie form of the i-KMS condition, Proposition 5.3.3. But ^ is separating for ^ and hence r^//2^ 7^ 0. -//2
Stability Therefore, the
o/ is well-defined and is F^ is invertible.
vector state
5.3.3. Next note that
(rD''
1 +
=
-'O
(P)
153
Equilibrium
i^-KMS
a
5] J^frf^i r ds,--- r" ./.T,,,
n^l
and
state
...
T,,,
by Proposition
(P)
JO
co and c/ are identical. Thus, the equivalence of t^-KMS normal state of co, and the factor property of CD
and hence the set of vector states of
uniqueness of of,
the
äs a
follows from Theorem 5.3.30
or Proposition 5.3.29. 0 and define D^^ and O^ by the 31^ such that \\PL P|| perturbation expansion given in the theorem. It follows from the estimates of (1) that 0 for all ^ G ^ and ^ G IR by Pro ||Q^^ Q^ll ^ 0. Moreover, ||Tf^(.4) rf (.4)|| position 5.4.1. Therefore, of^ converges to a state of and this state is a t^-KMS state
Next choose
^/,
e
-
by Proposition
5.3.25. co
By the
and of
same
approximation procedure one concludes that the uniqueness-factoriality equivalence
identical and the
are
before.
äs
It remains to
First, define
Q^(Z)
-
identify ^ by
Q^(z)
and
e^^+'^^/^Q.
Q^(z)
Q +
/
^z n
then
-^
-
vector States of
follows
^
-
>
^51
-'-1/2
l
< ^1 <<
5<
...
dSn^is^.(P]
'
T,,,,(P)Q
,
0
is
holomorphic for Re z G (0, 1) and strongly continuous for Re z G [0, 1] Q^. Now let D denote the corollary of the estimates of (1). Moreover, ^^(1) Union, over the intervals / C R, of the ranges ofE(I) where E is the spectral family of äs a
=
/f + P. If (^ G >
one
has
(^(^-^^^/^cp, Q)
=
(cp, (/(z))
for pure imaginary z. But both sides are holomorphic for Re z G (0, 1) and con tinuous for Re z G [0, 1]. Hence they are equal for Re z G [0, 1] by the edge of the
wedge theorem, Proposition
QGD(e(^+^)/2),
5.3.6. Since D is
Q^
(3)
Again
a core
for
e^^+^^/^,
one
concludes that
and
we
=
choose PL G
Q^(l)
=
^(^+^)/2n
^^ such that ||Pz,||
<
.
\\P\\
and
\\PL
-
P\\
0.
-^
Setting
/^,^(z)-co(^T,(P,)...T,,(P^)) one
has
\fA,L(z)-fA,M(z)\
<
Y.\O^(AX,(PL)"'^.J(PL-PM]"'^.APM]]\
.
7=1
But if SUp
Qi
G
^T, repeated application of the three-line theorem gives
|co(^T,(ß)---^z,(ßl))|
zD5_"J <
SUp l
=
sup
SUp|w(^T,(ß)...T,._/(ßy)...T,,_/(Öi))| ^6 K"
sup|a)(T,Xöy)---t(Ö,)^tJß).-.t,.,,(ß;+,))|
l
<
|M||fT||ß,-|| ,Vi
154
States in
Quantum Statistical
where the second step
Mechanics
the i-KMS condition. Combination of these estimates
uses
/A^L is a Cauchy sequence uniformly on D^"^j and thus the limit /^ is holomorphic in D*^j| and continuous on its closure '^_\. Moreover,
establishes that function
l/, (z) l
sup
IM II ||P||"
<
.
zeT)^'| But the truncated functions
truncated functions and if
finite linear combinations of
products of the nonappropriate combination of the /^ it holomorphy and continuity properties and the desired
are
define F^
we
follows that FA has the correct
the
äs
,
boundary value. To derive the uniform bound
C0r(^,
T,, -
This
equality follows
(P),
sup|F^(z)|< Now recall that
a
.
,
T.,,, .
.
FA
first take P
we
(P), T,,-KP),
.
,
T,
.
.
.
^H^ and remark that
G
T,_KP))
,
(P), A, TjP),
.
.
.
,
T,^.,,
(P))
from the definition of the cor in terms of äs
.
co
and the i-KMS
above that
sup|cor(T,XP),...,TjP),^,TjP),...,Tr,.,(P))|
sup I
.
COKT/,(P),
condition. It then follows
,g-j,(")
-
on
.
<j
function F
over
the Index set
3 and its truncation FT
are
related
by
F(/)=Fr(/)+
Y.
FT(J]F(I\J)
,
aeJc/
where
a
is any
for
<
n
l/l
point
in /. Next
where n>2 and
obtains from the
|/|
assume
that
Tn\{(Tn\Y^ ^^ ^
m=\
<
apply
l
by
2\^\-\\I\
-
1)!
\I\=
n+ l
one
with
||^/||
< 1.
^
(^
-m+
1)!
2"-'"
+ ^
'
+
this result to truncated functions
\o}(AiA2
>
<
2"/7!|l/8 e^/--l|
^)|
< l
and
|cor(^/)|
|cor(^,,...,^)| for all
|Fr(/)|
H-5;]"C,_,2-i(m-l)!
<
-
We may
and
< l
foregoing equation
\FT(I]\
One has
\F(I]\
denotes the number of points in /. For
<
097(^1,^25
l, |cor(-^/, Aj}\
<
< 2.
5^)
Hence
2"-^(^-l)!
induction.
Combination of the conclusions of the two
preceding paragraphs immediately
gives
sup|F^(z)|<2"!||P|nM|| z^^^:] for P G ^IT; and the
general
result follows
by uniform approximation. Thus,
we
have
established the existence and convergence properties of the perturbation series given in the fourth Statement and it remains to identify this series with co^ when 2||P|| < 1. Let P G
^, and define
co^(^) by o}^(A)
=
(^(A)
and
and
Stability
[
>^(A)=
ds,---dSn CO(^T,-,, (P)
T,-
...
155
Equilibrium
(P))
.
J-\ -l <5i <S\ <-<5<0
It follows that
Y,o^:(A]
o,(AY^_:)
=
.
n>0 7>0
But rP G
r^
^T and it satisfies the analytically continued co-cycle relations -p
T,(r^,)
riH-,-r^,(rfj, i(s t} +
^
(r^)-'
(r!,)*,
=
(r^,)*.
Therefore,
ü;(^r^,)
=
w(^r^,/,T_,y2(r!,y2))
=
co(r5/2^r!,y2)
=
w''(A)co(r%, -i/2 r^,) -i/2
a;^(^)co(r!,)
=
^
.
These two identities lead to the relation
X:A":(^) >
But it follows from the
=
a,^-''(^)^;X(i)
0
>
of co^
representation
.
0
given
in
(2)
that A G IR
H^>
co^-^ has
a
power-series expansion
o,'^(A)
=
^r&^(A) n>Q
with
a nonzero
comparison
radius of convergence. Thus, one obtains
by multiplication
and
term-by-term
of power series
o^(A) ä^,(A) + Y^'^'' a,:(^)=(ö^(^)+)jä,:_x^)r(i) =
.
r=\ r-
In
particular
äs
(^)
cog
=
<^(^)
one
finds /.o
d)f(^)-cof(^)-co(^K(l)Now for /
=
{/,,
.
.
.
,
/ dswT(A,iis(P]] J-\
/l } adopt the notation
CDT(A; /)
-
cor(^, T,,,.^ (P),
.
.
.
,
T,,,^
(P))
co(^;/)-co(^T,v (P)...T,,,JP)) We
assume
l
r
<
77
-
(*)one
.
ds,--'dsnWT(A',{l, 2,...,r})
>'-l <^l
from
,
that
^^(A)= for
.
l and then
has
use
<-<Sn<^
the relation
(*)
to prove that it is valid for
r
=
n.
First,
States in
156
Quantum Statistical Mechanics
CO^(^) =O}^,(A)
+
/
V
dSn-r+l. ..dSnCOT(A] {
-
T
+ l
r=\'^-'^<Sn-r+\<---<Sn
/
X
ds\
J-\ But
-=
^n(^}
variable
=
.
.
.
,n
-
r})
.
,
/7})
.
{!,...,}
relation between
finds
one
l
+
dsi-
J-l <^|
where /
.dsn-r0}('^j{l,
.
.
<si <...<.y-,-<0
by change of Integration
^r(^)
.
.
,
and the
dsnY,^T(A: J)a;(1;
<---<5<0
sum
is
over
^COT(A] J)co(1;
,
the strict subsets of /. But the
function and its truncations
a
I\J]
J^l
/\J)
=
general
gives
co(A', I]
-
COT(A', I]
Jci
and hence
^n(^}
=
-l
^n(^)
dsi--- dSn{co(A', I)
J-1 <Si
=
/ ./-l
-
CO^^; /)}
<-<Sn
dsi 'dSnCÜT(A]I)
.
<si <---<Sn
This establishes the correct identification of the
perturbation series for P G ^I^. For general P the series is obtained by uniform approximation. Thus, the proof of the C*-version of Theorem 5.4.4 is complete. The W^*-version can be established by the following Variation of the above arguments. The first Statement of the theorem is established by an inductive reasoning which necessitates
multivariable version of the three-line theorem. This allows
a
one
to
Of for all selfadjoint P G ^l. Next one proves that if PM P in the strong Of strongly. This is an imaginary time version of the operator topology, then Q.^'^^ last Statement of Corollary 5.4.2 which follows from a double approximation procedure. One first introduces regularizations. define
^
-^
/T
PM:.
=
r
dte~^^'^~i,(PM]
\V
,
717
and PX, of PM and P, and then 2.5.22.
Next,
one uses
the
PM, explicit form A
^
PM and P^
of the
P strongly by Proposition regularization and a change of vari ^
ables to deduce that lim A->00
HA-'-PM. ;A--'-'"PM,;.
A^'-^/'M. ; -
for all
z
G
^_i/9,
and
a
A-PMA^-'-^'-PA/
similar result for P;, and P. But
one can
the uniform boundedness of convergent sequences, the vergence theorem, and the regularization that lim M
-^
HA-PM,; A--'--'PM,A
A-"'--^^PM,;O
-
A^'-^^PM^II
-
0
also conclude from
Lebesgue-dominated
A-P; A--'--P;.
-
A-''-"^P;P||
-
con-
0
00
for all
z
G
D_|
79.
It
now
follows from these two estimates and the uniform bound
edness property of the first part of the theorem that lim M
-^00
||A""PMA-^'-'~"^"PM
-
A'-'-'-PMÜ
-
^"P^-^'^-P
A-''~^'-Pn||
=
0
.
Stability
and
Equilibrium
157
Another
application of boundedness and the Lebesgue-dominated covergence theH^. But the i-analytic elements finally yields the desired convergence Q.^-'"' form a strongly dense subspace of ^l, by Proposition 2.5.22, and hence the continuity of Q^ allows the further discussion to proceed through approximation with analytic orem
>
elements in the
same manner
used for the C*-version of the theoreni.
Note that the vectors Q^ are cyclic and separating for ^( (or for 51^' C*-case) äs a consequence of the i-KMS condition, Corollary 5.3.9. then r^ G ^, and P G ^
(i^ by
use
of the
=
r!^./,Q
in the But if
r!^./4T_,/4(r^/4)o
=
analytically continued co-cycle (r^ } T/^i^i _/^;
relation. Moreover,
(r^ v^ (i _j^)
T-
Therefore,
Q^
=
=
_
r!,/,A'/Y^,/,o r-_,,,A'/^,v4(r:,y4)^ AV2pP *Q -//4^ ^ -i/4 ^"^
pP
-^
where J is the modular
conjugation
_
pP
A
TT-P
Q
-//4^i -i/4^^
5
associated with the
pair (^, Q). This corresponding to (^, Q), Defi nition 2.5.25, and this conclusion extends to all P P* G ^I by strong ap proximation of P by analytic elements and the continuity of P^^Q^. Thus, establishes that Q.^ Hes in the natural
cone
^
=
Q^/||O^||
normahzed representative of the i^-KMS state c// ^, Theorem 2.5.31. This allows one to readily identify the modular operator associated with co^. For example, if A exp{//} is the is the
contained in the
unique cone
=
modular operator associated with
(^, ^)
and if
Ap-e^ + ^-^^^ then
by
a
perturbation expansion
tf (^) for all ^ G ^l and
(^, Q.^).
,
^ G
=
A^^A-'^
IR, i.e., Ap implements the modular group of the pair product formula, Corollary 3.1.31, shows that
But the Trotter
A^iA=
lim
(A^'^/" e'^^/' Ve'^^/V)>
.
n-^oo
This
representation demonstrates that Ap ^ operator for the pair (^, Q^) by Corollary
C
.:^ and hence
AP is the modular
2.5.32.
The conclusions of Theorem 5.4.4 extend to
(T. ^)-KMS states, for any ß e U,by rescaling. Thus, for each (T, j5)-KMS state CD the theorem provides a recipe for construction of a (T^, ^)-KMS vector state co^. The following corollary summarizes the properties of this map c/jt-^ co^.
158
States in
5.4.5.
Corollary p
=
Quantum Statistical Mechanics
State
^l, ß^R, cü^ by
be
(^, T)
Lei
p* G
C*-
a
associate mth each
p
"^^)=
or
W-dynamical System
(T, ß]-KMS
state CD a
and
for
(T^, ß)-KMS
(0^4^4)00 (Q^Q'^)
and rß/2 /p/^
(i-
=
ao
+ ^.
l)"/
rs, rs\
J^l
JO
n> l
r^n-i
/ J52---/Jo JO
Js7r(T(P)---T,,,(/'))aa
.
y^^co^-^co^ is an isomorphism of the set o/ (T, ß)(T^, ß)-KMS states which maps extremal points inverse map is given by (yf)"^ y^^.
It follows that the map KMS States onto the set of into extremal
points. The
The map yf is well-defined, and it maps extremal points into extremal points, by Theorems 5.4.4 and 5.3.30. Next assume that P G ^T, then Q^ nf,)(r^ßj^)Q(^. But the image of co^ under y~p^ is then determined by the vector PROOF.
=
rß/^
_P
ü~
7I,Xr^V2)^- + E / 70
=
rs\
JO
> l
It follows
readily from the analytic continuation, that
Ü"'' Thus,
y'/'y^co
=
establishes that
co
and
(yf )~^
fSn-i Jo
second identification of F^ in
7l,,((rf^/2)" Vco(rr^/2)ao
-
interchanging
=
dSn7l,,(T^,,^ (P)
ds2---
ds,
7"^^
follows
and
by approximation with Of tinuity property Q^"
a
yf
T
is
and T^ an
sequence
one
=
^0
-
-
-
tf,, (P}r^ß/2}^co
Proposition 5.4.1,
and
.
also has
y^y'/'co^
=
<jf This
isomorphism. The result for general P of analytic elements and use of the con-
-^
.
It should be
emphasized that the map y^ is not affine. "time-dependent" approach to the stability of KMS states which is less complete than the above but is of interest for various reasons. First, it emphasizes different physical aspects; it is dynamical rather than kinematical. Second, it plays a fundamental role in the subsequent attempts to justify the KMS condition äs a characteristic of equilibrium. Third, it applies to ground states and ceiling states for which the above "time-independent" analysis has no analogue because the whole modular structure is lacking. To illustrate the difference between the two approaches and to explain the time-dependent time-independent nomenclature we first recall the discussion of the finite-dimensional System which preceded Theorem 5.4.4. In this example the relation between the perturbed state co^ and the unperturbed state co arose by expansion of the co^ density matrix in terms of the co density matrix. But these density matrices determine the probability distribution of the various kinematic states of the System and hence this method is basically a time-in dependent, or kinematic, way of linking the appropriate equilibrium disThere is
a
second
Stability
and
Equilibrium
159
approach is distinct in that it is time-dependent and identify the perturbed equilibrium state äs the evolute of the unperturbed state under the perturbed dynamics. Conversely, the ori ginal State should arise by evolution of the perturbed state under the original dynamics. This latter phenomenon, which is often referred to äs return to equilibrium, is not universal but depends intimately upon some form of ergodicity of the System (^, T) and purity of the state co. It appears that the necessary ergodicity should be some form of asymptotic abelianness. Properties of this type for the dynamical group correspond to dispersion, with time, of local disturbances. Their role in the time-dependent approach is illustrated by the following result which establishes the tendency of perturbed KMS states to return to equilibrium.
tributions. The second
consists of attempting to
Proposition 5.4.6. Lei (M, T) be a C"^- dynamical System, where ^ has an identity, and T^ the perturbed evolution corresponding to the perturbation P* G a. Further let co^ be a (T^, ß)-KMS state for ß G [Ru{ 00} and CD P a weak^'limit point ofi^co^ äs t tends to infinity. If (^^ T) is asymptotically abelian in the norm sense, i.e., if =
T,(5)]||=0
\\mJ[A, for
all A, B e^l, then If
PROOF.
is
0, then of is
j?
(T, ß)-KMS
a
trace and
a
co
is
state.
a
trace, i.e.,
a
(T, 0)-KMS
state. Next
[f5\{0} and let d denote the generator of T. The generator of T^ is ^ + dp dp(A) /[P, A]. The (T, j5)-KMS condition now follows by verification of the
assume
where
CD
ß
G
auto-correlation lower bounds of Theorem 5.3.15.
Explicitly
one
has
(^^)-S^^Ä""'<"(-)-.M)-^:|:g!^ \im-iß(o''(t,(A*](5 + 5p](x,(A]))
<
t^OO
-tßcD(A^Ö(A})
=
+ lim
jßco^(T,(^*)[P, T,(^)])
/> oo
-ißcD(A^Ö(A))
-
The first step
uses
the lower
semi-continuity of
the auto-correlation lower bounds for
follows from
.
co^,
w,
asymptotic abelianness. Similarly, \iß
-i(D(A^d(A]]
=
lim
-i
v\-^u
=
+
log (u/v),
the second
uses
xd, and the fourth oo, i.e., co^ is a -f ground state,
the third relies
on
öi
=
co^(T,(^*)(5(T,(.4)))
/>oo
>
-
lim
O}^MA*)[P, T,(^)])
=
0
t^ oo
and
co
is
a T
ground
state. The
case
ß
oo
is identical.
proposition does not establish the existence of the limit of T*CO^ but if unique (T, jß)-KMS state, this follows because all limit points must be equal. In fact, one can deduce a stronger result for jS G IR by use of the timeindependent theory. This
there is
a
160
States in
Corollary and
Quantum Statistical Mechanics
5.4.7.
assume
that
Let
co^ be
(^, T)
is
an
extremal
asymptotically
(T^, ß]-KMS abelian in the
state
for ß
G
U\{Q}
norm sense.
It follows that the limit
(D(A] exists
for
PROOF.
all A
G
Corollary
state CD, of
Example
co^,
^, and
is the
CD
of(it(A]]
lim
=
iiniqiie (T, ß]-KMS
5.4.5 establishes the existence of
and
co
is
factor state
a
by
an
vector state
extremal
of <jf
(T, ^)-KMS
.
vector
Theorem 5.3.30. But it then follows from
4.3.24 that
lim
coMT,(^)C)
=
co(AC)o}(B)
t-^oo
i.e., Ti,('it(B]] converges weakly
to
co(5)llf^
for all 5 G "jH. But of is
a
vector state
of
co
and hence
(ß(A]
=
lim
co^(Tt(A]]
.
t-^oo
Thus
asymptotic abelianness and extremality imply the existence of the limit äs the argument works equally well äs ^ oo one deduces that the two limits are equal. In fact, if one replaces the pointwise limit by a mean value one can deduce an existence result without extremality of co^. One exploits the mean ergodic theorem and the methods of Section 4.3. Again the mean over positive t and the mean over negative t are equal. In Order to fiirther develop the time-dependent formalism it is useful to introduce a stronger notion of asymptotic abelianness. States and
^
Definition 5.4.8. A
asymptotically
C*-dynamical System (^, T)
is defined to be
Z'(^o)-
abelian if
f J
dt\\[A, T,(5)]||<+D
c )
for all .4, B in the norm-dense This strong form of existence of the
*-subalgebra ^o-
asymptotic abelianness
is useful because it
limits 7.^ of T^^T+/, äs ^ ^ 00, sition 5.4.10. Thus, if ft}^ is a i^-invariant state, norm
lim
CO^MA)}=
r->oo
oj^(T^_,T,(A}}
lim
=
for all P G
implies
the
^o, Propo
co^(y^(A)}
/->oo
exists. The
verify
problem with such conditions, however, is that they are difficult to particular models. This is not surprising because these conditions a form of ergodicity. Nevertheless in the simplest example, the ideal
in
express
Fermi gas, the L^ property
can
be verified.
EXAMPLE 5.4.9. Let ^ be the CAR
subalgebra, i.e.,
the
algebra over L-(r) and 31^ the even C*subalgebra generated by even polynomials in the annihilation
and creation operators
a(f)
in Section 5.2.4. Thus
a(Utf),
and
T
and
is
a
a^(g).
Next let
group of
T
denote the free evolution discussed
Bogoliubov transformations, it(a(f))
=
Stability
(U,f](K) If
a(f]
V
>
3, define 5Io
and a* (g] with
äs
/
'^H^o)-asymptotically then
II [^, T^(ß)]||
=
the
/
(InY^'l^-
and
161
Equilibrium
rf>/(p)e'> + '-(''^-")'
*-subalgebra
of ^^ formed
by
even
polynomials
of the
and g of compact support. It follows that the pair (^g, T) is abelian. This is established by first noting that if ^, B ^ ^g,
is bounded
by
of terms of the form
a sum
UlV2 d^'x
\(UJ'](x)\~
.
This follows from the CARs because
have chosen
we
compact support property. To estimate these bounds
(U,f](x]
where
we
subalgebra
and the
note that
J dy f(y] J d'p ^P(^-y^^(p-^^^
=
(2nr
=
(-^mt]-'''^
have used the Fresnel
an even
we
l d'y f(y)e-^^''-y^''^^e-^^^
,
integral
f d'pe^^P'
=
(iu/t]''''
.
But then -ix~/4t -it^it
r ^
^ -^/ ^> ^^y~i^^f(y)^^yi^^.
(u,f](x]=(-471/0 Therefore,
\(u,f}(x]\ where
|A^|
<
(47tM)-'/2
is the volume of the support A' of
/ ffxg(x)(U,f](x)
d^'xg(K)(U,f)(x) <\\g\
v=
(iA'l
|A"|)'/^
(47:|,
Thus, the bound for ||[^, 1^(5)] || is integrable for all
l, 2, then (^e, '^} is
choice of ^o- If, however,
,
/. Hence,
A"
where A'' is the support of 0^. v>3. If
11/11 / dy !/(;,) l <J^1_ (47ik|)'^-
J^'
not
L'(^o)-asymptotically
redefines ^o
abelian with the above
the
algebra of even polynomials in a(f) and a* (g), where f,g are in the domain D(X~] of the multiplication operator, (X'^f](x) =x~f(x), and their Fourier transforms vanish in a neighborhood of the origin, the property is again valid. To see this one simply remarks that
.^(., ^,/)
=
one
(^)7'^>^^/')(^|:)^'-'^""" i\- /-./öl 2V
Now the
äs
7''KI;^)"^^^^^^^^^^'"
,i(p~-H}l
right-hand side can be bounded by the Cauchy-Schwarz inequality and use special properties of /and g, e.g., /vanishes in a neighborhood of the origin and df/dpj,d~f/dpj ^L'(R''). Thus, one finds \(g, UJ)\ < constant x t\~- and (^e,T) is once more L'(%)-asymptotically abelian. of the
162
States in
Quantum Statistical Mechanics
Note that the
integrability conditions
L"([R'). Thus,
if
asymptotically
abelian in
II [^, i(B)]\\
one
a
integrable appropriate test
built with
integrable
are
if ^ and B
arise
considers the füll CAR
are
slightly
intricate
more
whenever A,
5, is
or
the
sense.
an even
The commutator
element of
functions but the anti-commutator
a
norms
norms
subalgebra ^lo
\\{A^ i:(B]}\\
are
both odd elements.
The motivation for the introduction of
provided by
directly from the action of U on algebra ^, then (^, T) is L^(3Io)-
Z/^(^o)-^symptotic
abelianness is
following.
Proposition 5.4.10. Let (^, T) follo\vs that the limits
be
L^ (^Q)-asymptotically abelian C*-
an
sy Stern. It
y^(A)
T!,T,(^)
lim
=
r^ioo
P* G ^o- The y^^ are norm-preserving norm for all A ^^ and P *-morphisms of^ \vhich satisfy the intertwining relations exist in
=
7T-:/7 If^
has
an
identify,
then the
.
adjoints y^
are
affine transformations of the
E^i into E^n with the following proper ties\ 777^ 7^ map i^-invariant states into i-invariant states and extremal (1) i^ -invariant states into extremal i-invariant states.
States
(2)
(3)
7^ map (r^, ß]-KMS states into (T, ß^KMS and extremal (T^, ß)-KMS states into extremal (T, ß]-KMS states for all ^G(ff^u{oo})\{0}. ff ß ^ ^\{^} ^^^^^ f^^ maps 7^ coincide in restriction to the (T^, ß)The
KMS states.
PROOF.
By differentiation
and
integration
one
obtains the estimate
r ds\\[P,
||/T.(^)-/T(^)||<
T,(A]]\\
Jtl and hence
y^(A)=
/,T,(^)
lim r-^oo
exist in
norm
for all ^ G ^o ^nd P
=
P* G
2Io. But
äs
^o is
norm
dense and T^ and
norm-preserving the limits must exist, and satisfy ||7(^)|| ||^||, As limits of *-morphisms the 7^ are automatically *-morphisms and ning property follows because are
=
7(T,(^))=
lim
/,T,+,(^)-
5^00
lim
Tf/,_^T,+,(^)
=
T
for all ^ G ^. the intertwin-
-^f^il^)
S^00
Since the 7.^. are *-morphisms their adjoints are affine maps of the dual which map positive functionals into positive functionals. But 7(11) (1) and hence 7;^ must =
map
PsTi into E^I.
(1)
If co^ is
T^-invariant,
it follows that
and
Stability
(y>0(T,(^)) because of the
co^(7T,(^))
=
a3^(r^7^(A)}
=
163
Equilibrium
(y>0(^)
=
intertwining property. Moreover,
(7>0(^T,(5)) =^(7(^)Tf(7(5))) Thus, if cü^ is extremal i^-invariant and
M is
.
invariant
an
[R, then
mean over
M((y>0(^T(5))) =M(co^(yi(^)/(y(5)))) (co''(y^(A))w''(y^(B)) (y>0(/l)(ylo/)(5) .
=
=
where the second Step follows from the
Theorems 4.3.17 and 4.3.22. But
extremality given by
and hence this düster property is theorems. If co^ is
(2)
equivalent
to
CO^(T!,T,(^))
lim
is
co^(i,(A))
lim
-
/> 00
y^co^
(51, T) is asymptotically abelian extremality of y^co by the same
(T^, ß)-KMS
(7>0(^)= and
,
theorem and the criteria for
ergodic
mean
r>CXD
(T, j5)-KMS by Proposition for all ß G [R\{0}. Now
Corollary 5.4.7,
5.4.6. If co^ is extremal
if
^
+00 and
^
co^ is
CD
an
is extremal,
extremal
by
(i^.ß)-
i.e., an extremal T^ ground state, then co^ is pure by Theorem 5.3.37 and T^-invariant by Proposition 5.3.19. In particular, co^ is extremal i^-invariant and KMS State,
y^co^ State
is extremal r-invariant
5.3.37. The argument for
If co^ is
(3) where
co
is the
the maps
they
are
this is
The
7^
an
exactly
j5
=
extremal
the first Statement above. But
-
cxo
they
on
extremal
an
y^o)^
ground
T
is
state
state
for
vector state of
the extremal
points
ß
e
U, then
y^co^
co^, by Corollary
of the
(T^,j9)-KMS
=
5.4.7.
ground
Theorem
(T^, j?)-KMS
states
CD
=
f_cD^,
Therefore,
states and because
must coincide on the convex closure of these extremal
the set of
a i
by
is identical.
(T^, jß)-KMS
unique (T, ^)-KMS
coincide
affine
by
5.4.6 and hence it is
by Proposition
points.
But
the Krein-Milman theorem.
by
algebraic analogues of the wave operators, or in quantum-mechanical scattering theory and hence we refer to them äs Moller morphisms. They provide a global comparison of the two evolutions T and T^. These morphisms are not necessarily *-automorphisms because they may not be invertible. Their kernels are zero but their ranges may be strict subalgebras of ^. In scattering theory the formation of bound states is a typical phenomenon which signals the lack of invertibility of the M011er matrices and in statistical mechanics lack of invertibihty of the M011er morphisms y^ certainly occurs if the perturbation P isolates finite Subsystems. The formation of bound states, or the isolation of Subsystems, is indicated by the appearance of a point spectrum for the perturbed evolution. But if y^ or y_ is invertible, the intertwining relations give *-morphisms
y.j_
M011er matrices, which
are
occur
y^ry-i and hence the
point spectra
spectra differ the y^
are not
of
T
and T^
invertible.
=
/
are
identical.
Conversely,
if the
point
164
States in
Quantum Statistical Mechanics
Although the ranges of the y^ are not automatically equal to "^ the morphism and isometric properties imply that they are C*-subalgebras ^ and the intertwining property shows that these algebras are i^-invariant. It is then interesting to note that if (^, T) is asymptotically abelian in the norm sense, the (^_i_,T^) are asymptotically abehan in the same sense, e.g.,
||[7^(^),Tf(y^(5))]|| Similarly
if the
=
|h/^(M,T,(5)])||
^
||[^,T,(5)]||
.
is L ^(Ulo) -asymptotically abelian the
pair (^(,T)
L^(7(^o))-asymptotically
(^,T^)
are
abelian.
Thus, stability of the various forms of abelianness under asymptotic perturbations of the group is directly related to the invertibihty of the corresponding M011er morphisms. In fact, stabihty of the L^-condition is essentially equivalent to invertibility of y.^. (Assume (^,T), is I^(^o)-asymptotically abehan in order that y^ exist. If the y^ are invertible, ^ and (^, T^) is I^(y(%))-asymptotically ^ abelian. Conversely, if is the limits (^, T^) Z.^(7_j_(^o))-asymptotically abelian, =
T_,Tf(^)
lim
ß^(A}=
/>oo
exist for all ^ G ^ and P
=-
P* G
y(%)
and for those P
G
^^o
n
y(%)
one
has^i^yji.) EXAMPLE 5.4.1 1. Let ^ be the CAR algebra over L~(U'') and T the free evolution 0. Let P given in Example 5.4.9, with /z a*(f)a(f) be the quadratic perturbation of Example 5.4.3 and then T and T^^^ are both Bogoliubov transformations it(ci(f]] =
=
=
a(UJ) erators
and -
V^
rf(a(f))
a(Uf-^f).
The groups U and U^-^ have infinitesimal genand -V- + /lE" where E is the projection on /. It is a Standard result of
scattering theory (see
=
Notes and
that the strong limits
Remarks)
W^g=
U^:^U,g
lim t-^oo
exist
on
Z-(R').
Thus the M011er
Moreover, the W
are
morphisms
7^^ exist
on
^l and
y(a(f})
=
a(Wf).
given by
(W^(l](p}
=
j d^'qWi(p,q)(l(q]
,
where
W,(p,q)
=
l mp)f(q) -q^i &.(q,^^^
6(p-q)-J^^^>^^>_ p-
and
/'v A(^^)=i-^yj>If
A(0)
>
0, the W
are
unitary
and the 7^
\f(p}\' '^
-
are
p^--
invertible.
i.e., if
i/.>^<-, J
P~
i
,
If, however, A(0)
<
0,
Stability the W
are
Equilibrium
165
unitary. In this case V" + >!" has an eigenfunction /;. exactly the orthogonal complement of /;. more general quadratic perturbations and, for example, if
isometric but not
and the ranges of the W These results extend to
are
.
-(a*(VV)a(/) +a*(/)(VV)),
P=
and
then
Tf(a(/))=e-'-'(/.^^/)a(/) One of the most
significant
features of the
.
time-dependent
formahsm is the
"stability" conditions necessary for consistency with the time-independent formalism. The origin and nature of these conditions, which we set
of
readily understood by examination of an extremal Tpoints of view. The time-independent formalism expresses co äs a vector state of a unique i^-KMS state co^ which is related to co by the convergent perturbation expansion of Theorem 5.4.4. After rescaling this assumes the form
intend to derive below, is KMS State
co
from the two different
o)^(A)^oy(A)^Y.(-\Y rTi
l ds^rds^--'!" 7o
Now consider the orbit of
dependent co
J^cor(^,T,-,^(P),...,T,-,,(P))
.
JQ
-'O
under the
co
perturbed evolution T^. The timegeneral ergodicity hypotheses
formalism indicates that under certain
evolves into
a
t^-KMS
evolved state is the this definition of
state. It is natural to define the
unique i^-KMS
stability requires
co^(^)
vector state of co,
System
äs
stable if the
namely, co^. Explicitly
that
-
lim
co(Tf (^))
.
t-^ 00
It then follows from the t-invariance of
Proposition
co^(^)
given
in
co(T_,TfM))
lim
-
and the definition of T^
co
5.4.1 that
/-^oo
lim
-CO(^)+
^"^
r / dt, J-tr dt2'" J-t 5^f ^-t
J^.CO([TJP),[...[T,(P),^]]])
.
>1
seemingly different series expansions for co^ and the identity of these series, term by term, yields the stabihty conditions. The simplest and most important is the first-order condition. Thus
one
obtains two
Theorem 5.4.12.
System
(^, T)
and
Let
co
be
assume
a
(T, ßYKMS state
that lim
co
is
over
the C*-
or
strongly clustering, i.e.,
cD^(Ai:t(B))
=
CD(A](D(B]
t-^oo
for
all
A,
B e M.
It follows that
lim
/
'^OOJQ
dt(D([A, it(B]]]
=
i
l
JQ
ds(DT(A,iis(B]]
W-dynamical
166
States in
Quantum Statistical Mechanics
and hence r^
/ dt(D([A,it(B]]]^^
lim
T^ooJ_j^
forallA.B^^, PROOF.
For each ^,5 G ^ the function t^^(jo(Ai:t(B]] is the boundary value of a function, which we denote by z^-^ co(^T.(5)), analytic in the strip D/j;0
a;(^T,(5))|
Slip
<
IMIIII5II
.
zel^ß Now the
(T, j5)-KMS
condition
gives
co(Ai:fß(B))
=
CD(BA)
and therefore
r^
/ JQ
[T
dtcD(Ai:,(B]
-
i,(B)A)
=
/ Jo
dtco(Ai,(B}
Ai, + ,ß(B))
-
rß =
The second
allelepiped
identity
with
/ Jo
follows because the
corners
o}(Ai:T + is(B))
integral
co(AiT + is(B))
=
T> 00
'-^^7o
co(A)a}(B)
Lebesgue-dominated
The
gives
CO(^)CO(T/,.(^)).
convergence theorem that
.
the first Statement of the theorem and the second
two cases T
-^
00.
condition
/
special significance in ergodicity hypotheses
^G [Ru{oo}. Comparison of stability properties
the two
dtcD([A,,,(B)])=Q
J _j
will be of
Proposition
Uco(T)
JQ
lim
Section 5.4.2 where
it is
equivalent
demonstrate that under
we
to
the KMS condition for
perturbation formalisms
also
gives higher-order
for KMS states.
5.4.13.
mical System
=
+
l^CG
suitable
the par-
weak convergence of
///'" dscü(Ai:T is(B})= / dsco(A)a)(T:i,(B))
by subtracting the
stability
z^co(y4T.(5)) around (0,ß) is zero. But
rß
Combination of these results follows
of
and
equivalent to the Therefore,
rß
lim jm T-
.
(U,,(is/2)Ti,,(A^]Q.,,, U,,(T)U,,(is/2)n,,(B)^,,]
=
It then follows from the
ds(D(Ai^,(B)-ArT + is(B))
(0,0), (r,0), (T,ß),
at
and the strong clustering condition is to the projection on Q^^ äs T oo.
lim
i
(51, T)
Let
\vith
co
^
be G
a
(i:,ß)-KMS
R and
assume
state
that
co
the C*- or W*-dynastrongly clustering.
over
is
and
Stability
167
Equilibrium
It follows that
lim
r dt,---
lim
...
[T, (B], A]]])
-
Jtn-\
/
Hr
=
r dS2'-' r' dSnO^T(A^Tis,.(B],...,Ii,,(B]]
^^1
JQ
JQ
Jo
5e 21.
forallA,
Let 5 e
PROOF.
l" dtn co(K (B), [
7;,^oo7o
Ti-^ioo
^T and introduce B(s\
,Sj)
,...
T/^-^(5)
=
(5).
T/^,
We first argue
that
/""^O+i /
r"^^ya;([T,^.,.(5), Ch^.,,(^(.,,...,5,-))
^^i-"
JQ
Jo
Jtj
rß -i
=
for all C
r^j
/ Jo
e
^51...
by an application of the KMS condition and conproof of Theorem 5.4.12. The KMS condition allows
in the
äs
reexpress the left-hand side
to
one
CTT,,,(B(SI,...,SJ+I)))
-
^. This follows
integration
tour
/ dsj+iO}(Ci,j(B(si,...,sj+i)) JQ
the difference of two terms LI
äs
and ^2,
where rß
rTj+\
/
L,=
/ JQ
dtj+i
Jtj
rsj,\
/ JQ
J^i-..
r^y+i
L2 L2= -
/'"^^y+i / Jt,
/' '^5yco(CT,^.^,(5(^i,...,5y,0)))
^^i---
.
and
a
Sk + ß, k by a change of variables s'j, Sk+\ of of shift the contour integration subsequent =
rß
fT^j+i
L2
=
r^
l
dtj+i
dsi...
rß =
LI +ii
l
ds\
X
.
-
l one
,
...
l
,7
,
and
s'j
=
ß
-
si
finds
dsjO}(Ci:,^^,(B(ß-sj,si-sj,...,sj.^-Sj,Q)}')
f^j
.,,
l dsj+\ (
JQ
JQ
-
l'
JQ
JQ
Jti
,
JQ
Jo JQ
Next
dsjO}(C^^,(B(ß,si,...,Sj)))
{co(Citj(B(ß
-Sj+sj+i,..., sj-i
O}(CT:TJ+ (B(ß ,
-
sj + Sj+i,.
.
.
-
.sj^i
sj + sj+
-
sj -\-
1
,
sj+i)))
sj+i.sj+i)})}
.
1.2. A similar change of variable gives the desired identification of Z/i identity is valid for a general B E ^ by approximation with analytic elements äs in the proof of Theorem 5.4.4. Next define C and X by
Another
c(/i,...,o
=
X(T,,...,Tn)=
[Tj5),[...K,W,^]]]
l 'dt,
JQ
Application
of the above
,
r~ dt,,., r dtn03(C(t,,...,tn]) Jt\
identity
Jtn-l and strong
clustering gives
.
168
States in
Quantum Statistical Mechanics
lim TI,...,/,,
X(T,,...,T)
-^00
dt,w(c(t,^...,tj]] -E-Hr% 1^^^ r dt,...r^ r,,...,7;->oo Jo Jf._^ j^l rß
rs-j-\
ds\--' l
X
dSn-j W(B(SI
rß
+
n
>2 and foT
The we
n
.
dSi---
Sn-j]]
,
ds(D(AB(s^,...,Sn))
\
=
jo
one
obtains tlie result of Theorem 5.4.12.
proof is conducted by induction. Theorem
now
.
f^n-^
/
(-/)"
7o for
.
,
Jo
JQ
suppose the result is true for 7
=
5.4.12
1,2,...,-
l
gives
the n=^l
The above
.
case
identity
and
then
gives lim ri,...,r
>
X(T,,...,T) 00
=
-
E Jo/
(-^')"
^^1
'
~f rß
+ l
/
.
Jo
+
(-/)"
/
we
tween
a
the
have
dSn
/
(-/)"
changed
l"' ds
ds^
COT (A , T/,,,
(5)
,
-
.
.
,
T/,^_^.^ ,
(B) )
üj(^T
(5)
...
T;
(5))
Jo
ds,---
Jo
where
(^))
JO
Jo =
^"-l
rsn-i
/ ^^,,_y
X
/'"'"' ^^'^~J ^(^-n-, (^) Jo
["' dsn caT(A,
T,-,
(5),
.
Jo
.
.
,
T,
(5))
,
the
integration variables and used the general relation be(For a similar argument see the conclusion of Theorem 5.4.4.) This completes the induction and the proof.
function and its truncations.
proof
of
To conclude the discussion of
stability of KMS states we examine various properties of ground states, i.e., KMS states at the value +00. All oiir comments apply equally well to ceiling states. The principal difference between ground states and i-KMS states, in the present context, is that the ground states have a tendency to be less stable. There exist C*-systems (^, T) and perturbations P such that (^, T) and (^(, T^) have unique ground states which are disjoint. Thus these states, which are automatically pure by Theorem 5.3.37, generate unitarily inequivalent irreducible representations (see Example 5.4.15). We begin with a positive result concerning existence of ground states of perturbed C*-systems.
Proposition State
co
and
5.4.14.
Let
assiime
M has
It follo\vs, for each ground state.
P
(^, T) be a C^-dynamical an identity,
=
P* e ^, that the
System mth
a
ground
perturbed System (5l, T^) has
a
Stability The
PROOF.
Uoj(t}
=
ollary
ground
exp{zY//,J
state
is
co
7Cf^(^)",
e
5.4.2 establishes that
But Hfo +
i-invariant, the corresponding unitary representation > 0, by Proposition 5.3.19. Moreover Cor-
T^ is unitarily implemented by the operators +
7r,,(P))
is lower semi-bounded and hence
T[(Ü(P}
i/^
lim
^
S
||(//,,
such that H^j + +
may find
one
Tc,a(P)
>
7r,o(P)-;j^)iAJ|-0
a
A e R and
that cü^ is
The
a
+
ground
7r..(P) state
-
of
;j.7i,,(^)liAJ
>
a
AH^o but
.
Now let con be the sequence of vector states associated with the i/^ and co^ point of this sequence in the weak* topology. It follows from the estimate
(^,7i,,(^*)[//.o
169
Equilibrium
and HO,
U^(t) ^g'X//o. sequence of unit vectors
and
-|M||'||(//co
+
n,,(P)
-
a
limit
;:1,,)^J|
(51, T^).
following example
shows that co^ is not
necessarily jCco-normal.
algebra over L-(IR'), T the free evolution, perturbation discussed in Example 5.4.11. It follows from Example 5.3.20 that (^, T) has a unique ground state coo, the gauge-invariant quasifree state with the two-point function Let ^ be the CAR
EXAMPLE 5.4.15.
and P
=
ß*(/)fl(/)
the
coo(a*(ö'i)(^2))
^
te, Tgi)
,
where
(fg)(p)
=
g(p)^
ifp'^<^
=
0,
if p- > ^
.
But Example 5.4.11 establishes that the M011er morphisms perturbation }^P exist for all >l G (R and are invertible if
d"pp-V(p}\'
7.1.
corresponding
to
the
'
Therefore, the T and T^'^ ground states are in one-to-one correspondence, by Pro position 5.4.10, for small A, i.e., there exists a unique T''-^ ground state. This ground state is given by CÜQ^ y^^coQ and it is the gauge-invariant quasi-free state with twopoint function =
o4^(a^(g,)a(g2)) where
=
(W+g2.TW^gi)
W^ is the M011er matrix of
(g2. W^TW^g,]
=
,
5.4.11.
Example possibilities for COQ^ which depend upon the support properties of the Eourier transform /of /. If supp f ^ {p^ < {j}, then COQ is T''-^-invariant and CD^'^ O}Q. If supp / g {/?- < ;u}, the states are distinct but two cases occur which can be differentiated by examination of the particle density. The state coo has finite particle density because There
are
various
=
N^(coo)
=
^o,o(a^(fn)a(f)]=-^ ^p-<^ [ d^'p
n>l
^^^^
for any orthonormal basis {fn}n>\ of L^(A). Hence, COQ is locally normal with respect to the Fock representation. The particle density of CD'-^ is given by
170
States in Quantum Statistical Mechanics
^A(CO^O
/ d'p d'q u(P -q} Jl--
(271)-"
=
,
J
where
U(p)=
j d^'xe'P^ JA
and
^+(-, )
is the
integral
kernel of the M011er matrix
^+ given in Example 5.4.1 1. by explicit calculation. On the i^i} ^ 0, the integral for N^(co^'^) can diverge because of the sharp cutoff in the r-integral. But it follows from the special structure of quasifree states (see Notes and Remarks) that co^^^ is locally normal if, and only if, it has finite particle density. Thus, co^^^ is not locally normal and hence the two pure states co and CD^'^ must generate unitarily inequivalent representations. Now if supp fr\{p^ /(} other hand, if supp fr\{p~ =
=
0,
then
N\(o}^^)
< +
oo
=
The
foregoing example demonstrates that ground states are less stable than ß. There exist T ground states which do not have T^ ground states äs normal states. The phenomenon which occurs in this ex ample and destroys the normality is quite general and is usually called an infrared divergence, The energy difference between the perturbed and unperturbed Systems is finite because P is bounded, but this energy is shared by an infinite number of particles each with an infinitesimally small energy. A similar Situation occurs for the ideal Böse gas. All particles in the ground state have zero energy but a bounded perturbation can introduce an infinite number of infra-particles. Another aspect of instability of ground states is given by examination of the KMS states at finite values of
condition
fT
lim
/
dt
T^OGj_j^ In Theorem 5.4.12
we
0}([A, T,(5)])
showed that each
-
0
.
strongly clustering (T, /?)-KMS
state
co
satisfies this condition for all ^, 5 G ^ but for ground states this condition can fail for certain A and B; in particular for those A and B which give emphasis to
low-energy
excitations. If
co
is
a
strongly clustering
T
ground state, the
Fourier
transform of
teU^(i^,u^(t)i^)-(i^,Q^)(üa^,i^}
,
where Uco is the
unitary group implementing T on ^ and i/^ G ^^5 measures the distribution in the vector state il/. The next result demonstrates that the energy stability condition is satisfied if, and only if, these measures give zero weight to the
origin, i.e., if,
and
only if, infra-particles
Theorem 5.4.16.
are
absent.
Let (^, T) be a C*- or W -dynamical System, clustering T ground state, and U^o(f} exp {itH^o} the unitary implements T in the cyclic representcition (^, TT^O, ^co)=
co a
strongly
group which
Stability The follomng conditions
are
and
Equilibrium
17l
equivalent:
r
/
lim
(1) T
-^
00
J^co([^,T,(5)])
=
0
J_j^
forallA.B^^, fT
->ooJ_'P
T
/
lim
(3) T
for
all
-^
\l/
00
dt{(ß(Ai:t(B]]
0
Jr{(iA,^..(0^)-(iA, ^co)(0,, iA)}-0
^.
e
an s
>
0 such that
PROOF.
=
J_r
There exists
where
a)(A)(D(B)}
-
^.
forallA.Be
(4)
/
lim
(2)
denotes the spectrum
<j(Hco)
The state
{0}U[8, (^>
C
co
a;(^T,(5))
is -
of H^^.
and hence
automatically i-invariant, Proposition 5.3.19,
co(^)co(5)
(TI^(A''^^^, V^(t)Ti^(B}^^}
-
-(7ra;(^*)Qco,na,)(Qa.,7ra;(5)a.)
-
(1) trivially. To prove (1) ^ (3) it suffices to consider o; is pure by Theorem 5.3.37 and hence (S^j^^w) is irreducible. Therefore, it follows by an application of the Kadison transitivity theorem identical to the application in Example 4.3.24 that one may find ^ e ^ such 0- Therefore, that ^l^ü(A}^^^ ^ and 7rfü(^*)ßü
Therefore
\\i
(3)
such that
^
But
(2).
(i/^, Qfü)
=
(2)
^
0. Now
==
=
.T
.T
lim
/ dt(^,UJt]\l/]=
lim
/
dt
cD([A\x,(A)]]
=
^
.
T-^^J_T
r->oo7_7^ It remains to prove
(3) ^ (4).
(3) ^ (4): Let H be the restriction subspace R of ^ 0 C Q^),
of //^j to
a
closed, separable, Ua^ -invariant
y-oo
//
/
=
J^(;7)j.
7o the
spectral decomposition representation. Then
/
of //, and dv the
dt(ilj,^^^cp)=
i/^
e
in the
R. Hence condition
(3) implies
that
roo
lim
/ dv(p}g(p)2p~^ sinpT ^0
T^oojQ
corresponding spectral
r d(^,E(p]cp]2p-^^mpT JQ
J-T
for all (p,
measure
172
States in
for all g
functions p such that T
Quantum Statistical Mechanics
L\dv}. But the 2p~^ sin pT^ -^
>
oo.
It
uniform boundedness theorem then
uniformly follows immediately are
a(H) In
principle,
could
depend
C
bounded in that
[e, oo)
L'^(dv)
v([0, E))
0 for
=
implies
that the
for all sequences T^ some s
>
0, i.e.,
.
upon 5\ but
äs the closed subspace generated by a separable and ^^j-invariant it follows by an ad argument that we may choose s independent of the subspace. (3): Adopt the above notation. The strong clustering assumption means
countable number of such 5\ is still absLirdiim
(4)
=>
that
[dv(p)f(p}e^^P
lim r
for all
00
L^(dv). But if ^ G L^(dv), L^(dv} because 8 > 0. Hence,
/
also in
0
=
J
then the
function/defined by f(p) =g('p)/p
dv(p]g(p)2p~^ sinpT
0
=
is
.
Therefore, fT
/
lim r
for all
(p,\l/
G
,
-^
J_j^
00
dt(il/, e'^^>)-0
GCÜco-
The existence of
an
energy gap in the excitation spectrum,
i.e., the spectrum
of //fü, is not
generally expected for ground states of nonrelativistic Systems. Infra-particles are usually present because of a mechanism referred to äs Goldstone's theorem. Thus the stability condition, condition (1) of Theorem 5.4.16, is not generally expected for strongly clustering ground states. Nevertheless, this condition is satisfied for those ^, ^ G ^ which give no weight to the
energy, e.g., if
A, B e '2lQ([R\[-e, s]) for some spectral subspace of Definition 3.2.37, then
zero
g
>
0, where '^HJ denotes
fT
lim T
/
dt
^OGj_j^
M(Ait(B})
=
0
.
This follows which is
by noting that t ^-^ co(ATt(B)) is the boundary value of a function analytic in the upper half plane and the choice of ^ and B ensures that
this function decreases like
exp{ e
Im
z}.
Although the stability criterion for ground states is not valid on the whole C-algebra when infra-particles are present, it is possible that it holds on a dense C-subalgebra. This is the Situation if (^(, T) is Li(*i(o)-asymptotically abelian, Definition 5.4.8, äs we next demonstrate. Subsequently of such Systems with the aid of M011er morphisms.
Theorem 5.4.17.
dynamical System. It follows that
Let
C^ll, T)
be
an
we
discuss the
L^('^lQ)-asymptotically
stability
abelian
Stability and Equilibrium
173
)
dt
w([A, T,(5)])
=
0
)
for
all
A,
MO, andfor all (T, ß]-KMS
B G
(^, T)
and the
that ^ has
assume
Next
assume
Theorem 5.3.30,
continuity Therefore,
ß
an
(f, j5)-KMS
is
{ib oc}.
u
states
correspondence between of (^, f) with \\cb\<^\\
co
the
=
(T, ^)-KMS
1. Thus
we can
identity.
U and that
e
co
[R
G
=
one-to-one
a
jS
have
=
States of
mth
if necessary, and extend t to 5l CH + ^ by already noted, in the discussion preceding
First adjoin an identity, setting ff ((a, ^4)) (a, T/(^)). We
PROOF.
Theorem 5.3.30, that there is
states co
a
is
co
an
(T, ^)-KMS
extremal
factor state. But the
Therefore, by
state.
L^-asymptotic abelianness,
and the
of T, imply that (^, T) is asymptotically abelian in the norm sense. co is strongly clustering by Example 4.3.24. Thus, the stability condition
follows from Theorem 5.4.12. This condition combinations of extremal the condition may be
(T, ß)-KMS
expressed
states.
immediately extends to finite convex Using L^(^o)-asymptotic abehanness
äs
/
\
/-oo
/
co
dtlA, T,(5)]UO oo
and this relation extends to any Finally, consider the cases ß
(T, ^)-KMS
First for >0 define x, otherwise.
By Fourier transformation
As
oo.
=
by x,(l)
state
they
by
are
(In^^'/s
=
the Krein-Milman theorem.
similar
we assume
for l G
that
[s, 28]
ß
and
=
-{-
y^}
oo. =
^
has
one
r2E
X,(t)
=
~^
dAe^^-^
=
e^'^^/2(sin Bt/2)/(Etl2]
JE and hence
/im^Z.W-^1 Thus,
another
of
application
L^(^o)-asymptotic abelianness,
the
Lebesgue-dominated
and Fourier transformation
convergence
theorem,
gives
roo
/oo dto4[A,r,(A')])
^
00
=
/
lim
=
u
dt
x,(tM[A,r,(A^)])
J_Qo
l lim^0
r^^
/
J,
d(n,,(A*)Q^, ^^(;.)7r,,(^*)Q^)
>
0
for all A e ^Q. We have used EO^ to denote the spectral measure associated with the unitary group U(o which implements T and the second step follows from the ground state condition, i.e., the support of E^^ is in [0, cxo). An identical argument with /(/.) chosen to have support in [-28, -B] gives however roo
l
/ dt a}([A, T,(^*)j) 7-00
-
-
/.-
/ öf(7r(^)Q^, ^^0j-2 lim
-
E,,(}.)Tio,(A)^,,)
Therefore, ^00
/: for all ^ G ^0- The
general
dto}([A,it(A'')])
result follows
=
0
by polarization.
<
0
.
States in
174
Quantum Statistical Mechanics
We conclude this section with
discussion of
different type of
stability perturbed the System (^, T) satisfies a condition of asymptotic abelianness. Let (*5ll, T) be a C*-system which is asymptotically abelian in the norm sense, and let co be an extremal (T, jS)-KMS state for some ß e U. (If^ ß 0, we adopt the convention that co is an extremal invariant trace.) It follows from Theorem 5.4.4 and Corollary 5.4.7 that there exist perturbed states co^ of 5-1 for condition which is
a
a
in terms of
expressed
states and is often valid if
=
all P
=
P* e 51 such that:
CD^'^ is T^^^-invariant for P P'' e ^ and A sufficiently small. /l 1-^ 0)^^ is continuous at l 0 in the sense
(1) (2)
=
=
co^^^(A]
lim
^
=
^
CD(A] ^
A-.0
for all A The
(3)
^, and all P stability properties ^
co^^(it(A]]
lim ?>
hold for all ^ e In
^,
P
P* G 5.1.
=
=
a}(A)
00
=
fact, the co^^ exist for all i co"^^ co for /l
P* G
^, and
[R and
A
sufficiently
small.
Tüfo-normal for ß
^ U. (If ß 0, 'ü, the asymptotic abehanness is irrelevant, and property (1) follows because i^ and T are related by an inner co-cycle F^, Proposition 5.4.1.) one
may choose
G
=
G
are
[R and P
In the introduction to the next subsection
family
of states with
properties (l)-(3)
is
a
^
P* e
we
show that the existence of
stronger stability
a
condition than the
condition
/: occurring
dtoj([A, T,(5)])=0
in Theorem 5.4.17. It is not known if
a
version of Theorem 5.4.17 is
valid with the
stability properties (l)-(3) but some special cases can be handled. If O) is an extremal T-KMS State atvaluej5 G [R and co^ is the i^-KMS state given by Theorem 5.4.4, and (^, T) is asymptotically abelian, then it follows from Theorem 5.4.4 and Corollary 5.4.7 that the stability properties (l)-(3) are valid. If co is a ground state with an energy gap, i.e., the condition (^(H(o) ^ {0} u [e, oo) of Theorem 5.4.16 is fulfilled, then it follows easily from resolvent identities that H,o + ^T^co(P) has a unique (up to a phase) normalized ^ eigenvector \l/ of minimum energy for l small enough. One may verify that the corresponding vector states co^^'^ satisfy properties (l)-(3) and in this case the co^^ are still TToj-normal. There do, however, exist ground states co where properties (l)-(3) are valid but the co^^ are not Tito-normal. This is a consequence of the following proposition applied to the free Fermi gas. Proposition
5.4.18.
and
that
assiime
Let
(^21, T)
(^, T)
has
a
be
an
L^ (^Q}-asymptotica!ly abelian System,
unique ground
state co.
Stability
and
Equilibrium
175
follows that for each P P* G ^o there exists a ground state CD^ for (^, T^). If {cD^} is an arbitrary family of such ground state s, then //
=
lim
(D^^(A]
=
o}(A]
;. ^0
and
co^fi/M))
lim /
for
all A G
we
=
P* e
=
CD(A]
00
^o-
ground states of (^,T^) follows from Proposition 5.4.14 arbitrary ground state of the System. The assumed asymptotic that the M011er morphisms
The existence of
PROOF.
and
P
^,
->
let co^ be
abelianness
an
ensures
y1(A)=
/,T,(^)
lim / 00
exist for all P
between ^ and
=
P"" e
7(^l).
^o, Proposition 5.4.10, and the These
*-isomorphisms
7^, for all States
/
IR.
G
co^
=
also
defined
are
a}^oy^
by
define
the
*-isomorphisms intertwining relations
<7: (y(^),T^)
Hence, the C*-dynamical Systems
which
y
satisfy
Since
CD.
=
co^|.^^(gi)
have is
a
unique ground ground state, it
follows that p
co
p
o
y^
CD
=
and hence
CD(A)
=
CD^(y^^(A))=
lim
CO^(T,(^))
.
t'OC
Now
by
the estimate roo
\\yi'(A)-A\\ which is valid for ^4 G
<
m
/ Jo
dt\\[PMm\
^o it follows that
.limj|7f(^)-^||=0 for all A e '^.
Combining
this with
'^(yf (^)) one
o}{A)
=
concludes that lim /.
Thus
properties (l)-(3)
are
^
CD^-^(A)
=
CD(A)
.
0
valid.
Note that the discussion of the
preceding paragraph
is
applicable
to the free
Fermi gas if one works with the even C*-algebra ^^ over L"([R^') (uniqueness of the ground state follows from Example 5.3.20) and hence properties (l)-(3) are
valid for this model. Nevertheless,
Example
5.4.15 demonstrates that the
176
States in Quantum Statistical Mechanics
corresponding co^
are not TTcü-normal, and the absence of an energy gap in the of that the state cannot /f^o spectrum implies ground satisfy the stability properties of Theorem 5.4.16. The latter fact can also be shown by an explicit
calculation
on
elements A
ficiently singular
5.4.2.
Stability
In the
=
a*(/)a*(öf),
B
at the Fermi surface p"
where
/and
g
are
suf-
/i.
and the KMS Condition
previous subsection
we
studied various
States and our next aim is to establish that
State is
a(f]a(g)
=
=
stability properties of KMS stability of a state can imply that the
KMS state.
Roughly one deduces that all extremal r-invariant states asymptotically dynamical Systems which are stable under local of the perturbations dynamics are necessarily KMS states at some value [R In G u jS fact, no completely satisfactory result of this kind occurs {dzoo}. a
of
abelian
one has Z^-asymptotic abelianness. We will, however, prove two theoin this direction. The first presupposes stronger clustering properties than implied by extremality but relatively weak requirements of asymptotic
unless rems
are
abelianness. The second theorem is but with Let
L^-asymptotic begin
with
under weak
proved
abelianness of
clustering assumptions
(51, T).
description of the stability properties which are central properties arise from a reconciliation of two distinct of envisaging perturbations of a System, the time independent and the ways time dependent, in a manner already partially described in Section 5.4.1. First, assume that CD is to describe a state of equilibrium of the System (^, T) then it is physically reasonable to postulate that co is r-invariant, i.e., the values of observations are stationary in time. Now consider a local perturbation P of the dynamical group. One could well expect the perturbed System to have a T^invariant equilibrium state co^, which differs from o; in a substantial way only in the immediate neighborhood of the localization region of P, and such that the set of states co^^^ for A G [-, e] approximate co in the sense (D^^(A) (Jo(A) 0 for all vi G ^. This is the time-independent point of view. Second, äs |/l| envisage the evolution of the perturbed states CD""^ under the dynamical group T. If the group T is sufficiently ergodic, then one would expect the spatial effects of redistribution caused by the perturbation P to gradually disperse and the states US
a
to both results. These
-^
-^
co^^^ should evolve
to the state co,
lim \t\
for all .4
entiating
M. This is the
G
and
lim
integrating
one
>
i.e.,
one
(D'^^(it(A]]
=
lim T
-^
00
of view. But then differ-
obtains
r dt ß;^^([P,T,(^)]) lim
0}(A)
time-dependent point
>cx)J_j^
T
=
00
r
=
should have
i{o^^^ (IT(A))
-
^
00
/^ A|;^ft)^^(Ti^,(/l)) j _Y
Aul
cü^-^(T_r(^))}/i
=
0
Stability
by use of the r^-invariance mally, one then finds
of CD^ and the
and
177
Equilibrium
For-
time-dependent postulate.
/-oo
/oo
dt
^^
CX)
/ dt
lim
=
=
0
the time-independent postulate, and the exchange of limit and integral would, for example, be justified for P, ^4 e 9lo if (91, T) is L^ (^o)-asymptotically abelian. Thus, one concludes that the equilibrium state co of the dyna-
by
mical System
(91, T)
should
satisfy
f J
the
stability requirement
dtCD([P, T,(^)])-0
c
for all P, ^ G 21, or at least for a sufüciently strictly local elements of a quasi-local algebra. The
(91, T),
foregoing or
of
large
subset of P and A^ e.g.,
heuristic discussion relies upon a property of ergodicity of which we did not specify exactly. We postulated that the
(91, T, co),
suffidently dispersive, or mixing, to ensure that the perturbed states equilibrium. This problem has already been discussed in Section 5.4.1 and Propositions 5.4.6 and 5.4.10, and some form of asymptotic abelianness is certainly sufficient. But it also appears necessary to postulate that CD is relatively pure in order to ensure uniqueness of the limits of Ö}''-^(T/(^)) äs dl 00. Therefore, this discussion of the behavior of equilibrium states co is f based on four types of postulates: System
was
return to
-^
(1) (2) (3) (4)
T-invariance, i.e., stationarity in time,
ergodicity of (91, T), e.g., asymptotic abelianness, relative purity of co, e.g., extremality of co among the stability under perturbations.
These four
postulates
lead to the
r-invariant states,
quantitative expression
fT
/ dto}([A,Tt(B)])=Q
lim
T^<^J-T for ^, 5
G
91, of stability and
KMS State. so
suprising
(The
our
aim is to show that
relation between
when
one
the KMS condition at
zero energy in the form The first result of this type is the following.
Theorem 5.4.19. over
co
is
a
Let
(9J, T)
be
a
C*-
or
given
in
Proposition 5.3.14.)
W -dynamical System and
co a
state
91. Assume that lim |/|>CX)
for all, A, B G 91, and following conditions: (a)
they also imply that
and the KMS property is in fact not remarks that the above stability condition reexpresses
stability
co
is i-invariant.
all
\l/
G
||7r,([T,(^),5])iA||-0
assume
that
co
satisfies
the
178
States in
Quantum Statistical Mechanics
The fiin ctions
(b)
Co(Plt(A]]
t^
^K-^sup
CD(P]CD(A]\
-
,
(ß(Pii:,(P2)i:t(Ai^s(A2)))
-
o^(P\^s(P2)]o^(Aii:s(A2)]\
seU
are
L^-functions for
dense) (c)
co
is Stahle in the
all
A\^ P\^ ^2, f^nd PI ^-subalgebra 5Io.
i-invariant
in
norm
a
dense
(a-\veak!y
sense
fT
/r dt (D([A,it(B]])
lim r_oo r->oo
for
all A, B e ^o-
It follows t hat
PROOF.
=0
J_j
(jo
is
a
(T, ß)~KMS
By elementary reasoning
s täte
for
some
j8G[Ru{cxo}.
assume that ^o is invariant under regproperties of o; adopted in the hypotheses of the theorem are preserved under this regularization since the convolution of two I^-functions is an L^-function. For example, the stability condition gives one
may
ularization with Z/^-functions. The düster
J
dt
co([ATf(B)^i,(C)])
=
=
CG^IO and/GL^(ff^). Next, for P/, AI e ^o and i
jj j
dt
ds
dsf(s]o,([A^,(B), T,(C)])
f(s]
j
dt
co([^T,(5), T,(C)])
=
0
if y4, 5,
1,2 define FA.,p^
=
for
t G
F^,pXO
=
(^(Pi^t(Ai])
G^,P,(0
=
o^MAi)Pi}
-
-
and
G^.^p. by
co(Pi}co(A^)
,
oj(Ai)w(Pf)
[R. The i-KMS condition
can be reexpressed in terms Proposition 5.3.14. We derive concerning these functions.
forms of these functions series of observations
OBSERVATION 1.
äs
t
in
^FA,,P,(t]FA,^P,(t]
-
of the Fourier transthe condition from
GA,^p,(t]GA,^P,(t]
is
an
a
L'-function
and
/ dt{F^,, PROOF.
By
the
P,
(t}FA,, P,(t)
stabihty condition
one
-
GA
P,
(t)GA,, P,(t)}
=
0
has
f-OO
0=1
rf?Co([P,T,(/'2),T,(/l,T.(A))]) )
dt{FA^r^(A2),P\i:,(P2)(^}
~
<^.4|T,(^2),
PiT.lPz)
(0}
D
But it follows from the second
the
integrand is dominated by clustering assumption
clustering assumption and the i-invariance of co an L^-function uniformly in s. Now from the
that first
and
Stability lim
CO(PTX^))
=
Equilibrium
179
co(P)a}(A)
s>- oo
for A, P
G
ßut
^0-
lim
using
the
abelianness
asymptotic
a)(PiTX^2)T,(^iTX^2)))
lim
=
5> 00
one
also has
(/'.[iX/':), t,(^,)]T,+.(^2))
S^ 00
+ lim
Co(PiT,(^i)T,(P2T,(^2)))
s^00
CD(Pir,(A,))co(P2i:,(A2))
=
From this and similar relations and the
applied
to
the above
Lebesgue-dominated
in the limit
integral
.
5
>
convergence theorem it follows that
oo,
roo
0
=
/ J
Ö?^{CO(P,T,(^1))CO(P2TK^2))
-
Co(T,(^i)Pi)co(T,(^2)P2)}
-00
roo
/
=
dt{FA,,P,(t)F,,^P,(t)
-
G^p, (OG,p,(0}
J 00 /.QO
+
/
co(P2)w(^2)
^^(^^,,p, (0
GA.^P, W)
-
o/ -00
/oc dt(FA,^P,(t]
GA,^P,(t)]
-
-00
)
-f rf;{F4p,
WF^P,(0
-
GAP,(t)GAPm
.
J-c
Now, all the functions F^,,p, and G^.^p. hence their Fourier transforms
ishing
F^,,p,
and
are
G^.,P.
Z,'-functions, by assumption, exist
continuous functions
äs
and van-
infinity.
at
OBSERVATION 2
FA,,P,(P]FA.,P.(-P] for all /> G R
PROOF.
==
Since
[R and all
^/,P/
G
=
GA,^P,(P]GA._^P.(-P)
5Io.
F^,,p,, G^.,P. GZ,^nL
^
^^,
it follows
by
Parseval's theorem and
Observation l that /DO
/oo ^P^A,,P,(P)FA,,P,(-P)= l ^
^
00
If/is
an
L^-function,
^V(^2),P2W
one
^
=0)(P2J f(s]l,^,(A2]) j j dsf(s)FA,^P,(s t] ds
-
+
ds
one
Frf(A,],P,(p]=FA,^P,(p]f(-p] one
f(s)oj(A2}cD(P2)
.
the Fourier transform of this convolution
Similarly,
.
computes that
=
Taking
^
dpGA,^p,(p]GA,_^P,(-p]
J 00
obtains
.
derives
G^f(A.],p.(p]
=
GA,,p2(p}f(-p)
.
180
States in
Quantum Statistical Mechanics
these relations in the earlier
Inserting
relation, with ^2 replaced by T:f(Ä2},
one
obtains fOO
/OO dpF,,,P,(p)F^,,P,(-p)f(p]= / As
and
FA^^P^
measure
dpGA,,P,(p)GA,,p,(-p)f(p]
J OO
OO
ö/i,,p,
are
continuous in p it follows
by letting /
converge to the
point
that
öp
FA,,P,(P]FA.^P._(-P]
GA,^P,(P]GA,,P._(-P]
=
-
Now, introduce the unitary group
U,(t) implementing
T
in the
+
00}
in
or
e^^"'"
=
l
e^^P dE,,(p)
cyclic representation (^^, 71^0,^0^) associated with
OBSERVATION 3.
[0,
=
If
(-00, 0],
the
FA,P(P)J^O PROOF.
of
spectmm
is
H,,
then for any p ^ U there exists
a
contained
not
pair ^,
co.
P e
in
^lo such that
.
Let ^?/ C R be the set of j9 6 (R such that F^,p(/?) / 0 for and two-point clustering, one has
some
A,
P
e'ÜQ.
By asymptotic abelianness
/"^
l
/ dta)(AT:f(B)C)=co(AC)co(B}
lim T-^oc
for all
2T
.
J_'p
A.B.C^'^Q. Hence, it follows essentially cr(t/oj) is additive,
äs
in Theorem 4.3.33 that the
Spectrum
(j(U,] Now E(o
'fo({0})
=
FA^p(p]dp The
=
is the
+
(j(U,,) C(T(U,,)
projection
onto
then
implies
that
4.3.34. But it follows froni
in R. We argue that '^?/
=
[R
by
-
cr(^fo)
(j(Uay] by Example
CQf,j and hence
d(n,,(p-'^)Ü,,E,(p)n,,(A)^,,)
continuity of F^^p
=
(n,,(F')Q.,,,E,,Ti,,(A]Q.,,]ö(p]dp cannot contain
the limit
5^00.
=
isolated
points. Hence,
this, and Proposition 3.2.40, that p^^^l/., i.e..
+
+
.411^(^2)
It follows from the
and P
=
Q
=
P\'Cs(P2)
in this relation and take
clustering assumption
and the
dte-^^^^'FA,^P,(t}FA,^P,(t)
oX/^2)to(P2)|^/e-'^"^F,,,p,(0 oX^i)oXPi) l e-'>"^F,,,p,(0 Jdt e-WF,,^p, (OF^p,(0 dt
f/ is dense
contradiction. Assume that
doniinated covergence theorem that
0-y
.
K
l dtFA^p(t)e-'P^^ for all ^,P G ^lo- Next set A
.
=
Lebesgue-
and
Stability Replacement
of ^i
e
s
U. But
+
t}F^,,P,(t)=0
Fourier transformation in
by
FA,^P,(-P)FA,^P,(PO for
all
R.
p
F^2 P2(po
+
181
by T:-S(A\) gives
/dte-^^^^F^^^P^(-s for all
Equilibrium
if
Now,
p) 7^ 0,
/? + /?o ^ ^^
+
we
then obtains
s one
P)
=
^
find
can
^2
and
such
p2,
that
and hence
F^P,(p)-0 for all /? e /?o
'^. But the latter set is dense and
-
everywhere and
zero
CO(PT,(^)) for all P, ^ G ^0 and
ticular, dim ^
=
Spectrum of Ua^ is
ß
G
[Ru{oo}
t G
l and
F^i,p,
is
one
co(P)o}(A)
=-
implies
that
co
is
a
i-invariant character. In parassumption that the
^ for all /. This contradicts the
sided.
(Incidentally,
co
is
a
t-KMS
state
for all
case.)
OBSERVATION 4. tinuous
IR. This
t/cü(0
not
in this
is continuous; hence
/^4,,p,
conclude that
we
If
positive function
ö-(//^ü)
(|)
on
is not
one
sided, then there exists
a
unique
con
(R such that
FA^P(P}
^(P}GAAP}
=
for all p G PROOF. that
If /> G R is
F^,,p, (-p) ^
fixed, there exist, by Observation 3, elements ^i, PI
G
^o such
0. Define
^(
)^|WZ^
.
FA^.P.(-P} Then
FAAP}
=
^(P}GA^P(P}
for all
^,PG^O by Observation 2. One can show äs in Observation 3 that 0 for some A,P e^ and hence O is uniquely determined by the last relation. Furthermore, äs ^/i,,p, is continuous we can use P^,.p, and GA^,P^ to define O in a neighborhood of /?, and hence CD is continuous.
GA,P(P} ^
It follows
immediately
from the definition of
FAAt}=f^P^A^(-f} Hence, FA^A'^ and GA, A-
are
real for ^ G
FA^P(-P) Therefore,
0(0)
and
GA^^P^(t)
GA^P
that
.
^o and
=
GP^A(P)
.
O is real and satisfies the relation
=
FA^P
=
l
by
a
direct
=
application of
a.(p)-'.
the
stability
condition. Thus ^ is
positive.
182
States in
We If
Quantum Statistical Mechanics
prepared
are now
a(U^^]
assume
is
from
one
sided,
now on
for the end of the is
co
that
FAA^}
=
is not
GA,p(t}
=
that the
one
by Proposition
projection
onto
5.3.19. Thus
^ P dEa)(p]
sided. If //co
is the
we
spectral
CQf,j, then
j ^"^(^*ao, (dE,,(p) E,,ö(p)dp)AÜ,,) j ^''^(^*QC., (dE,,(-p] E^Ö(p)dp)P^,,) -
,
-
foY A,P e^Q. Here, and in the
implies
KMS state at 00
a
(j(Ufj,)
of //^j and E^^ is the
decomposition
of Theorem 5.4.19.
proof
spectral
sequel, we identify ^ and nco(^). Observation 4 then absolutely continuous with respect to Lebesgue
measures are
measure, and
(P'n<, (dE,(p) Since O is
E,ö(p}dp)AQ,,)
-
positive
with
0(0)
Hence, 2Ion,
(p*n,/in,) Now
7r,o('älo) ( //,)''' is
is
.
.
and
(
=
dense in
strongly*
E5(p]dp]PQ.,]
((D(p)'/-^*n,rf(-p)(I>(p)'/n)
=
D((D(-//)'/'-)
C
-
l, it follows that
=
(P''Q.,dE(p]AQ.,)
(D(;7)(^*n, (dE^-p)
=
9JJtu
7t(o(2I)", by
=
.
Theorem
2.4.11, and
äs
closed operator it follows from the above relation that
a
aRncD(
by continuity
(5Q,^Q,,)
(a)(-//,)'/-^*Q,(D(-//,J^/Va,)
=
for all ^, 5 e 931oj. From this it follows that
AQ.,
=
^^
A'Q.,^
0
=
for all A G 9Jff;j. Hence Q(^ is
separating for ^l^^ by Lemma 5.3.8. Since 9J^fy^fo is a dense, C/f,j -invariant, subset of ^^ which is closed under regularization with functions in
L\
it follows
for
core
by
the
^(-H,oY^^.
reasoning But
used in the
93Iaj^oj is also
operator associated with the pair
A,
B e
identify
A'/-,
of
(^*Q,^*Q,,)
=
5.3.10 that
SfJff^Qoj is
a
where A is the modular
äs
=
(A'/-^au,A'%Q,,)
^M,, it follows from the remarks before Lemma 5.2.13 that
(D(-//)'/^ Now
core
(QU^j,^)- Moreover,
((D(-//,,)^/-.iQ,,ö[)(-//,,)^/-5Q,,) for all
proof of Theorem
a
T
with its normal extension to
ff,(A) be the modular
=
automorphism
=
A'/^
.
9JIf,j, and let
A^'^V^A-'^
group. As
co
is
i-invariant, it follows from Proposi
tion 5.3.33 that Itffs
for all t^s G IR. But
äs co
is
separating
=
(^s^t
for '^M^o
one
has
Stability inf \o}(AB'C) 5'6Co(rK(5))'
for all A, B, C ^ remark
imply
cr(//(ü)
and there
=
Equilibrium
183
0
^Jl^j by Theorem 4.3.23. Now Theorem 4.3.33 and the subsequent joint spectrum of H^^ and log A is additive. Since
that the
log ^(-H^] and
(D(B)cD(AC)\
-
and
=
log
A
[R by Observation 3, it follows that the function exists, a ^ e IR such that :=
^(p]
=
e-^P
p
H-^
log O(/?)
is linear,
.
But then
e-ß^Hence
Remark.
is
o)
t-KMS state at value
a
^
=
äs a
A
.
consequence of Theorem 5.3.10.
Instead of the
clustering conditions imposed on co in Theorem 5.4.19, require that the truncated -point functions or^ satisfy strong clustering properties for /i 2, 3, 4, i.e., if ^i ,^ G ^o there exists a non negative, nonincreasing, integrable, function gA^ -An on [R+ such that could
one
=
,
-
|cOr^K(.4i),...,Tf(A))| <ö'^,...^(sup \ti-tj\\ ^
for
/7
=
We
5.4.19
2,3,4 and
tt G R.
now
show that the
can
be weakened if
lianness. In
fact,
KMS States of
the
(T, 0)-KMS course, this
States, i.e., the discrepancy is
Let
stable factor states. The
(^, T)
be
a
sense
=
C*-dynamical System which of Definition 5.4.8, and co a
is
Either lim \ti-tj\^
Co(T^,(^i)T^,(v42)T^3(^3))
(D(Ai]o)(A2]cD(A^]
=
CG
for all AI, AI, A-^ G ^, or co is a factor co satisfies the stability requirement
state.
'"
l' J
for
-C
all A, P G ^o-
dt(o([P,x,(A}])
=
0
L^(^Q)-
i-invariant
^. Assume that
M
(2)
äs
0 because the extremal ^ extremal traces, are not necessarily factor states. Of absent if ^ has a unique trace-state.
abelian in the
asymptotically
(1)
abehan Systems occurs if
in this characterization
Theorem 5.4.20.
state over
clustering properties of the state imposed in Theorem one imposes a stronger form of asymptotic abefollowing theorem essentially characterizes extremal (T, ß)-
L^-asymptotically
only discrepancy
^
ij
184
States in
that
Itfollows
As
PROOF.
Quantum Statistical Mechanics is
(D
t^-^i:t(B)
L'(^)-asymptotic
an
is
extremal
(T, ß)-KMS
uniformly continuous
state
for
some
for each B G
^
G
ff^
u
{i oo}.
^o, it follows from
the
abelianness that lim
||[^,T,(5)]|HO
?^oo
for all ^, 5 G
^0,
from
4.3.34 that
Example
and hence for all
lim |r/-^-|>oo
inf '
for all
5 G ^. Thus if
^,
Cü(lt, (A\]--' T^(^n))
=
co
^(^l)
is
a
factor state it follows
^(^Az)
/;'
Z+ and AI G ^. Therefore, it suffices to prove the theorem under the three-point clustering assumption. If {Ey} is an approximate identity on ^, it follows from Proposition 2.3.11 that G
lim Tito ('a)^co
=
^co
a
and
this
using
one
easily
proves the
lim
two-point
(D(Ait(B]]
=
düster property
a}(A)co(B}
t^00
from the
5.4.19, Let
we
corresponding three-point property. Also, äs in the proof of Theorem may assume that ^o is closed under regularization by L^-functions.
US
define
for^/, P,-
G
FM
=
GM
=
F^,,P,(0
GA.^P^
=
=
o}(Pt^,(Ai)) ^MAi)Pi)
-
-
w(Pi}cD(Ai)
,
cD(Ai}(D(Pi}
,
%.
OBSERVATION l
.
t
^Fi(t)F2(t)
-
Gi(t)G2(t)
/oo dt{Fi(t)F2(t)
-
is
an
L^ -function, and
Gi(t)G2(t)}
=
0
.
00
PROOF.
By
the
stability requirement
we
have
/oo dta)([Pn:,(P2),it(Aii:s(A2W=Ii(s}-\-l2(s)+l3(s}-^l4(s} oo
where
/oo J;a)(P,[T,(P2),T,(^,)]T,+,(^2)) h(s)=
/
dto}(Pir,(At)[T,(P2),T:.,+,(A2)])
,
/"rf?Co([P,,T,(^,)]T,+i(^2)T.(P2))
,
J
Ij(s)=
,
J
-
00
OO
/oo rf;Co(T,(^l)[P|,T,+,,(^2)]T.(/'2)) -
00
.
,
Stability The 5-,
and
Equilibrium
185
integrands of /2 and /3 are dominated by L^-functions which are independent of using the two-point clustering and the Lebesgue-dominated convergence
and
theorem
we
lim
obtain
{/2(^) +73(5)}
/"
-
dt
7-00
^-"^
co(PiT,(^i))co([P2, ^t(A2)]]
/oo rf/ü;([P,,T,(^,)])cü(T,(^2)/'2) -co
r
=
J
where the last Step to show that
uses
the
dt{FA,,P,(t]FA,,P,(t]
stability lim
a
change
GApMGA^,pm
I\(s)
condition. Hence it is
=
5> OO
But, by
-
,
00
lim
74(5)
=
0
enough for Observation
l
.
5> OO
of variable,
/OO dt(a(-c^,(P^)[P2,r,(Ai)]T:,+,(A2))
.
-00
Again
integrand is dominated by the L^-function t i-^||Pi|| ||[P2,'z^/(^i)]|| iMill, independent of s, and the limit of the integrand äs 5 co is
the
which is
>
CO(P1)CO([P2,T,(^1)])CO(^2) by the three-point düster property. Thus, by theorem,
the
Lebesgue-dominated
/OO dtco([P2^,,(A,)])
=
Q
convergence
.
.00
One shows that lim^^ 0074(5)
=
0
by
a
similar
reasoning,
and this ends the
Observation 1.
Now, from the relations
FAA^}
=
Gp^A(-t],
F,^(^),p(0 -F^,p(^ + 5)
,
and Observation l it follows that
j dt{F, (t]G2(s -t]~G, (t]F,_(s Define
77^^p
=
F^^p
GA^P
and set 77]
H,(t] Then 77i is
an
7,^-function,
=
=:
77^,, Pj
F,(t]
-
-
0}
-
0
.
so
G,(t)
.
and from the relation above
j dt{H, (t]G2(s -t]
-G, (t)H2(s -t)}
j dt{F, (t]H2(s -t]
-H,
we
=
obtain
0
.
and
If
(7^(0
f^'^^ dEaj(p}
is the
(t]F2(s
-
0}
spectral decomposition
/^/""/^^P,o
V/
=
V^,.,/>.
-
0
of t/o), let
proof
of
States in
186
be the
Quantum Statistical Mechanics defined
measures
by
rfft(p)
=
(P;a, dE,(p)A,Q^)
dVi(p)
=
(A*Ü,dE,(-p)P:Q.,)-o4A^)w(P,)S(p)dp
w(P,)w(A,)5(p)dp
-
,
.
Then
FM
=
l e"''dß,(p),
G,(t)
=
y e"''dv,(p)
,
and thus
di.ii(p}
-
dvi(p)
=
Hi(p]dp
,
where /// is the Fourier transform of ///. From the convolution relations above now
we
obtain
OBSERVATION 2
H2(p)dv,(p)=H,(p)dv2(p}
,
H2(p]d^i,(p]=H,(p]d^i,(p] for
all^/,P/
Ulo.
G
Next, define
a
S
siibset 6' C [R
=
{p
^
OBSERVATION 3.
PROOF.
,
by
[R; HA^P(P) / 0
5 is
an
S is open since each
for
open set, S
HA^P
=
A,P
some
-S and
3Io}
^
.
0^5*.
is the Fourier transform of
an
L^-function,
and
thus is continuous. The symmetry follows from the relation
dpHA,p(p]
and
HA^P(Q)
=
0
=
(P'^Ü,,,dEcD(p}AÜ,,}
=
-dpHp^A(-p)
by this
OBSERVATION 4.
same
relation
There exists
a
-
(A'Ü,,,dE,,(-p)P^,,)
,
(or stability). well-defined
pair
/.(,
v
of cr-finite
measures on
S
such that
dl-^A,p(p]
=
HA^p(p]di.L(p],
dvA,p(p}
=
f^A.,p(p)dv(p)
for p ^ S, and, furthermore
^/^(p)
PROOF.
and
äs
{S\, 82,
=
-dv(-p]
Since S is open, S is the union of an increasing sequence of compact sets, HA^P is continuous, it follows that there exists a countable partition
each
}
of 5 into Borel sets, and elements ^/,P/ G % such that
fep,(/^)l>l for p G Si. Now, define
and
Stability d^^ p(p]
.
f dLi(p) ,
^
dvA,.p,(p}
dv(p)
=
,
^
ffAp,(p) Then /.i and for j!? G 5/, z l, 2, and Observation 2 implies that
^M^, p(p)
==
well-defined a-finite Borel
ffA, p(p]d^(p],
^0 and all
for all /4, P e
HA.,P,{P)
v are
=
p e S.
dvA, p(p)
=
)
-Hp.,A,(-p)
Now, the Radon-Nikodym theorem implies that there exist |/i|
Since
=
^i(p)d(M
^2(p)d(\^\
=
have we d(\i^\ + |v|)(j9) di.i(-p) -dv(p) _j^,,(p). Define subsets So,Soo, and 5/ of [R by
|v|-measurable
=
+
M)(p}
d(\^\
|v|)(-;?)
+
and
=
SQ
{P
=
e
iAi(/?)
=
0
if,
and
^oo-i/?
5;iAi(/^)-0}, Sf
Since
ö^v(;?)
v|)(;7),
+
=
j/^j(-jr,)
+
S such that
on
^KP)
5,
Finally,
//,,,P,(P) i/^^-
measures on
p(p)dv(p)
HA.
,,(^)^1^..,M^)^'^:^.,^.(-P)^_^,( functions
187
Equilibrium
=
R\(SQuS^)
only if, \IJ2(~p]
0^
e
5:>2(;^)
=
0}
,
.
deduce
we
immediately
OBSERVATION 5
SQ in the
sense
Define
a
=
that the sets differ
Borel function cp
Soo,
only
on
Sf
on a
set of
|;u|
|v|-measure
+
zero.
IR\^oo by ^^ rMp) ^2(P}
9(p)
Sf
=
on
S\S^
on
U\S
=
[l,
.
OBSERVATION 6
d^^A,p(p} on
|/i|
[R\*S'oo. Furthermore, (p(p} |v|-measure zero).
>
=
0 and
9(p)d^A,p(p) (p(-p)
=
(p(p}~^
Sf (except
on
for
a
set
of
+
PROOF.
It follows from Observation 4 that
|v|)(p)
,
dVAAp]=^2(p]HAAp]d(\ß\-^ ^\)(p)
,
d^iA,p(p)
=
^|^^(p]HAAp]d(\^\
+
for p e S, while
d^A,p(p]
-
dvA,p(p]
=
HA^p(p)dp
=
0
.
for p G 5^ As d^LA.^A(p] > 0 and dvA^.A(p) > 0 for all ;? ^ 0 and all A G ^o. it follows that (p (p) > 0 and since 0 for p e Sf. Thus
(p(/?)
=
(p(-p)~^
is
a
consequence
of\l/^(p)
=
-\l/2(-p]-
188
States in Quantum Statistical Mechanics
OBSERVATION 7.
If ^cü is the
projection-valued
measure
corresponding
to
H^^,
then
^o.(^o)-0 and, consequently,
E(S^^Sf]=^ PROOF.
.
From the relation
d^iA,p(P^
=
^l',(p]HA,p(p]d(\A
+
v|)(p)
it follows that
A^/i,p(^o) and
hence,
äs
O^iSo by
-dllA,
P G
0
-
/i^,p(^o)
Observation 3,
(P*Q,^,,(5o)^Q) for
=
=
0
%.
OBSERVATION 8
(P*ao,^(^/Ma.) for all y^, p e 911,, PROOF.
=
(9(-Ho.]"'^'^o^. (^(-//,,)^/-PO,,)
-^
7r^(5I)^
By Observation 6,
has
one
(P*Q, dE,,(p]AÜ,,]
=
d^i^^p(p)
-
(p(p)dvA,p(p)
+
co(P)co(^)^(;7)J;7 +
a;(P)co(^)(5(/?)^/7
-((^(;7)'/^^*a.,^^.(-/^)c/)(;7)^/'PO^) for p G over
IR\-S'co, and ^,P G ^o, where we have identified ^l IR\iS'oo Sf u SQ^ we obtain from spectral theory that
and
7rfo(3I). Integrating
=
^oOro
C
/)(9(-//,,)'%,,(-(5y- U^o)))
and
(p*ao, ^oX^/M^o.)
where
we
=
(P*a.o,^.o(^/ u^oMa.)
-
((^(-//,,)'/-^*Q,,(p(-//,,)^/-pQ,,)
have used the relations
ECO(SQ)
and
Efo(-(Sf^SQ))
=
Eaj(Sf
u
Soo)
which
follow from Observations 5 and 7. Since 9Io is to all
strongly*
dense in SPl^^
by Corollary 2.4.15,
^, P 9^0..
Next define
Ef Then
/
e
aJI.
=
[aR;nj
.
this last
equation
extends
Stability
and
189
Equilibrium
OBSERVATION 9
E^(Sf]
=
E^(Sf]Ef
=
.
Since
PROOF.
^itH^,ffl'^^-itH^ for all
EfE^(Sf]
r G
[R, it follows that Ef
^'^
-
e'^"^ and hence
commutes with
,
E^(Sf]Ef=EfE^(Sf) By Observation 8,
.
has
one
(E^(Sf](^ -Ef}P*Ü^,E^(Sf)(^ -Ef]A^,,] =
(cp(-H^]^'^A^(^-Ef]ü^,cp(-H^]^f^P(\
-/)Q,)
-0
for all ^, P E ^[Rcü, and hence
eo(^/)(l-^/)-0 on
Now, ßcü is cyclic and separating for the Ef9)f^ by Lemma 4.3.13. Define HE
modular operator and modular Then
=
j
Neumann
algebra 9Ji Ef^^Ef EfHo) and let A and / be the conjugation associated with the pair (93l,Q).
=
HE
.
p
dEE(p]
=
von
Hf^Ef
j
p
dE^(p)Ef
is the spectral decomposition of //. As e'^^^ leaves QCÜ invariant morphism group of SOI^, it follows äs in the proof of (3) ^ (2) in
JHEJ
=
JEE(B]J
=
=
=
and defines
an
auto-
Theorem 3.2.61 that
-HE
and hence
for all Borel sets 5 C [R. In
Ef
=
JEfJ
particular,
=
JEE(Sf
u
EE(-B]
it follows from Observation 7 that
^oo)^
-
EE(-(Sf
u
5-00))
.
But
^^(-^oo) by Observations
5 and 7.
EE(SQ)
=
0
Thus,
Ef and Observation 9
=
=
EE(-Sf)=E^(Sf}Ef
implies:
OBSERVATION 10
Ef=E^(Sf)
.
Now, Theorem 4.3.22 implies that E^^ 'ö>({0}) is a one-dimensional projection, äs QÜJ is separating for ^E it follows from Theorem 4.3.23 that =
and
inf
5'CO(TM(5))'
IcoiAB'C)
-
a}(B]o}(AC)\
=
0
190
States in
Quantum Statistical Mechanics
for all A, B, C e QJl^. As HE and A commute
strongly by Proposition 5.3.33, (2), it subsequent remark that the joint spectrum E of subset of IR". But since üoj is separating for 9[R,
follows from Theorem 4.3.33 and the
(log A,//^) Lemma
is
a
closed additive
3.2.42, (2), implies that Z is Symmetrie, and
OBSERVATION 1l.
The
joint spectrum
2 of
we
conclude
is
(log A,//^)
a
closed
subgroup
of [R-. We next show
OBSERVATION 12.
(J(HE)
=
is not
E,o(Sf)
one-dimensional,
then
Sf
is dense in [R and
^.
PROOF. ff (HE) is a group by the remarks before Observation 1l, and äs 0 is a simple eigenvalue of H(o it follows from the assumption and Eco(Sf} Ef that (J(HE] i=^ {0}. But cr(//) cannot have any nonzero isolated points because this would imply that H^^ has a nonzero eigenvalue /l with a corresponding eigenvector i// such that =
c/^(OiA-^"''A But then
(lA,7ü,.(T,(5))a) for all 5
e
^.
=
(U,,(-t)llj,7l,,(B)ü^)
Therefore, choosing
-
e'^^^(lA, 71^(5)0,,)
B such that
(iA,7i,(5)Q^)^0 this contradicts the fact that lim
n,o(i:t(B))
=
a)(B)^
?>CXD
in the weak
topology.
It follows from
Example 4.3.34 that
(T(HE)
=
U
.
OBSERVATION 13
(T(HE) PROOF.
This is demonstrated
^
äs
(J(H^-E)
Q
ff(H^-E)
in the last part of the
OBSERVATION 14. The restrictions of the are
absolutely
continuous with respect to
.
PROOF.
,
,
""^-"(^^
=
dv.Ap}
=
proof
measures
Lebesgue
di^i^p
ro,
ps,
pes^,
-"^'"^P^'^P^
P ^
^
0,
p ^
S^
This follows from the relation
and
measure, and
U.XPV;., {
of Theorem 5.3.22.
dvA,p
to
^o^^oo
Stability
d^iA,p(p} with the
together
Next, define subsets S^ -
Then S
are
191
HA^p(p]dp
=
d^^AAp')
^^ P^ '^o,
=
C [R
=
0' P^S^
by
n (p'^^^^^^^(p^ A 6
Equilibrium
equations
d^^A^P^
^+
dvA^p (P]
-
and
0}'
^
^-
n {p''^^AA^(p^
=
A 6
^y.Q
0}
<
^^to
closed sets, and since
U
^-
{P-.HA^A^P]^^}
.
^3lo
by polarization,
it follows that
5+ But the
measures
dp.A,A*(p]
^^^
5_
n
5
n
dvA,A^(p]
0
=
.
non-negative
are
for p
7^ 0,
and hence it
follows from Observation 14 that
we
may
assume
measure zero.
that the inclusions
are
sets from
By subtracting the latter
^oc and 5o
strict and hence
SQ
SQO ^ S^^ where the bar denotes closure.
C 5_
5*0
^00 Q -^+5 except for sets of spectral
Therefore,
C S-
one
,
has
OBSERVATION 15
^n'SQnS CS+nS-r^S Now, by Observations 7 and 10,
one
E,,(S^)
=
0
.
has
=
^
-Ef
and thus
OBSERVATION 16
O(H^,E]^'S^ We
now
finish the
Case 1.
proof
Ea)(Sf)
.
of Theorem 5.4.20. We consider two
is not one-dimensional. In this =
=
hence S^r\SQC^S
=
=
servation 15. It follows
0, and we may Ea}(Soo} 1^1 + v l -measure zero.
assume
=
=
[R
IR
=
by by
IR
Observation
Observation 5, and
=
^, Now
that ^^^
=
E,^(Sf} 5o
:^
1
by
Observation 10, thus
0 by modifying
^\,^i
It follows from Observation 8 that
(A'/-^*^^,A'/2pQ,J
by
Observation 13 and
since 6*00 C 5, and hence this contradicts Ob
S. But
S^0 that Ef
^(HE)
case
\. If not, cr(//i _;) show, ad absiirdum, that Ef hence 6*00 [R by Observation 16. But then SQ S^ 12. We
cases:
=
(P*ß^,^a,)
-
((^(-//^)^/-^*a(p(-/f,,)'/^pa,,)
on sets
of
192
States in
for all y4, P
Quantum Statistical Mechanics
y)l(o- As 501^0^ is
6
A
A^/^,
for
a core
Proposition 5.3.33, (2), it follows presentation of //,^ and A that
and H^o and
A^/"
(p(-H,,)
=
strongly, by joint spectral re-
commute
from this relation and
a
.
It follows that
EC{(\og(cp(-p)),p)-p^U} Now,
2 cannot have any isolated
and Z is
closed
a
Z must have I
(1) (2)
=
one
Z is
(3) In
case
points by reasoning used in Observation 12, U^ by Observation 1l. As (j(Hco) IR it follows that =
IR-, array of
equidistant straight lines, not parallel with the log A axis, orgin. straight line through the origin not coinciding with the log A axis.
an
of which contains the a
3, there exists
-ßH,,,
-
of
of the forms:
one
Z is
subgroup
e R
j5
a
such that Z
{(-ßp, p]]
=
/? ^
and thus
IR}
log
A
or
^
But Theorem 5.3.10 then
are now
Case l a.
two
HA^ A*
e-ß^'^
=
that
implies
the treatment of Case l,
complete There
.
the
we
.
is
co
must
r-KMS state at value
a
ß.
eliminate possibilities (1) and
Hence to
(2)
above.
possibilities: =
^ for all A
In this
e^Q.
case
co(^^*) =FA,A^(0)+a}(A)cD(A'')
GA,A^(0}^CD(A)co(A^)
=
for all A
^0, and hence
G
Case Ib. we
may
is
co
HA,A*(PQ) ^
is
co
is
(T,0)-KMS
a
^o and
cD(A'A)
some
-
d^^A,A^ (p]
=
HA,
A'
(p]dp
-
continuous,
&, PQ +
&), where
,
deduce that -
l)dvA,A^(p)
HA,A^(p)dp
=
(l-Cp(pr')d^A,A4p}=HA,A^(p}dp HA^A^ is
then
a
HA, A* (p)
real function >
0 for p
G
have two
we
(PQ
that
v
is
a
positive
measure on
(PQ
((p(p) this interval. It follows that
v -measure zero.
except for
a
(p such that
But
d/^(p)
=
=
-
-
> l on
(PQ
-
ß)-
We
e, ;?o +
s)
l)dv(p)
now
^^
ffA,A^(pQ}
>
0,
deduce from the relation
and
=
dp
(p(p) > l for p e (PQ (p(p}dv(p) and so (p(p) , PQ -^
.
ffA^A^(p)dv(p)
set of /.i-measure, and hence
(p(p)
,
possibilities: HA^A^(PO) <0-
e, /?o +
d^A,A^(p]
on
is
HA^A^
that />o / 0 and that HA^A^ (p) 7^ 0 for all /? G (PQ positive number. From Observation 6 and the relation
(q)(p]
As
state.
p^ G (R. Since
assume
d^^A^A^ (p) we
trace, i.e.,
a
0 for some^^ e
=
spectral
B).
a, po +
E) except (PQ
l for /? G measure, zero. Thus
But this
{(log((p(-p)),p)-pe
>
means
we
that the set
for
a
set of
, PQ +
e)
may choose
Stability
and
Equilibrium
does not contain any point of the form (d, p), where d <0 and p e As e is contained in this set, this excludes possibilities (1) and (2). The
=
T-invariant character, and hence then
E^(8^}
ad absurdum, that S
\-
=
S^Q^"^.
is
co
KMS state at all values
a
If not, then SQ
=
S^o
a(H^}CS^^{0}^U
i.e.,
co
has
cr(/4j)
contained in
one
isolated
no
of the sets
is KMS state at value +
e
=
!Ru{ oo}.
If
5^e _argue,
^ and hence
^ [R,
and
S^. n 6*0 n 5 by Observation 16
.
it follows from
points,
and thus
[0, +00),
ß
because 6*00 C
E,^(Sf) ^ 0_a.nd S^0
contradiction with Observation 15. Thus SQQ
j^0m
But since
s).
=
Eoj(Sf) + H, =
, PQ +
HA^A^(PQ) < 0 is treated by noting that HA^A*(-PO) -ffA,A*(po}H in this case, it follows that co is Ef.^(Sf] is one-dimensional. If J5'cü(5'/)
case
Case 2. a
(po
193
oo or
co
is
a
Example 4.3.34 that ö-(/4j) is ground state or a ceiling state,
oo.
By summarizing the results of the last two subsections, we obtain an almost completely satisfactory theory for the connection between stability and the KMS condition for C*-dynamical Systems (^,T) which are Z^(^o)-asymptotically abehan in the sense of Definition 5.4.8. Assume that 51 has an identity H, and let co be a t-stationary state on ^. If P P* G ^o, it follows from Pro position 5.4.10 that the M011er morphisms =
rf-
Ti^T,
lim /-^oo
exist
strongly
for A G IR.
Furthermore,
yf T.
has the
one
intertwining
relations
Tfyf
-
and lim
y^(A)
-
A
/i^o^ for all v4 G M
by
the estimate /i
OO
\\y^(A}-A\\<\^\
d\s\\\[P,r,(A)]\\ JQ
which is valid for A e^Q.
Now, there exists
a
unique
state
co^^
on
'y^(^) satisfying
(o^(y':^(A))=w(A) and co^ is
r'^-stationary by
the intertwining relations. But co^-^ extends to a by Proposition 2.3.24, and applying an invariant mean to this extension composed with T"^ we obtain a state co^ on ^ such that the relation above remains valid, and state of 21
(1)
0}^
is
1^ -stationäry.
Next, it follows from the estimate
|co^(^) that
-
c,(A]\
=
\co^(A
-
y>^(A])\
<
\\A
-
y^(A}\\
States in
194
/l
(2)
Quantum Statistical Mechanics
cü^^
\-^
A
is continiioiis /
0 in
=
co^^^(^)
lim
t he sense
-
co(^)
all Ae"^.
for
Now, define
co^^
state
a
W
on
by
<(^)=cünTi''(^)) From the relation
0,^^(^,(A))
CO^-^(Ti^T,(^))
=
it follows that
The limits
(3)
lim^_
o^^^^(^t(A]]
00
cD^^^(it(A)}=a}(A),
lim
family {co^^; (l)-(3) (including the
We call any
5.4.21.
Corollary
+
t-^
P
=
(^,T)
Let
P* G
be
an
co^^(T,(yl))
.
a
s^} of states satisfying requirefamily of pertiirbed states ofco.
(^Q)-asymptotically
an
=
cof (^)
-
<
(D^_f}
L^
mical System, and assume that 51 has P* e ^ÜQ, |/l| (9/^, a?id let {60^^; P
e% and
00
^a, |A|
existence of
all A
for
lim
/> -00
ments
exist
<
abelian C'-dynai-stationary state Sp} be a family ofpertiirbed states of
identity.
Let
co
be
a
Consider the follovving conditions:
(D.
(Iß)
cü
(2)
(a)
is
an
extremal i-KMS
state at
value
CD
has the
three-point
lim \titj\-^oo
ö;(T^,(.4i)T^,(^2)T/3(^3))
inf
ß.
düster property
=
co(A[}(D(A2)cD(A2}
i^j
(b)
(D
satisfies
the
stability property
\im\cD^^(A}-(D(A}\/l for
^
+
;.->o
'
^
=
Q
^''
all ^ G 51.
It follows that
for jß (2)(a)
G
is
(2) implies (Iß) for some ß G Ru{oo}. Conversely (Iß) ([R { oo})\{0} implies (2) and (Iß) for ß 0 implies (2) when replaced by the weaker düster property u
=
M(a}(Ai(B)))
=
CD(A)CD(B)
all
for
A^B G 5l, and any invariant mean M on U. particular, (Iß) for some ^ G IR u { 00} and (2) are eqiiivalent ifco is a factor state or z/ 5t has a iinique trace-state. Fiirthermore, the family of pertiirbations {co^^} can be chosen such that In
co"^ (A) (B)
(O
in the
if (l) (/(l)
following holds with
holds with
cases:
ß ß
e =
U -\-oo and there exist s
an
>
0 such that
Stability
(^(HCO) (C)
//(l) cases
holds mth
0}^
ß
can even
{0}U[,
-^oo, and (^, be taken to be
We first show that the
PROOF.
C
stability
T) a
+
and
00)
has a unique groundstate. i^^-KMS state at value ß.
condition
195
Equilibrium
(2)(b)
is
äquivalent
In these
to the
by
now
familiär condition
/oo dtw([P,T,(A)])
=
(*)
0
00
for all A^ P G ^0- But this is
a
consequence of the relation
r^5T^4([P,T,(^)])
Ti^T,(^):=^-Ü
JQ which
gives / 00
CD(A)
0)^-^(7^ (^))
=
:-
co^-^(^)
-
Ü
/ JQ
dsco^^([P,T,(A)])
/oo dsco^-^([P,T,(A)])
w'-^ and hence
(^)
-
w(A))/l
=
r dso,^([P, T,(A)])
-i
.
J -CG
The
Lebesgue-dominated convergence theorem and requirement (2) on the family now immediately imply that the two stability conditions are equivalent. Thus, it follows from Theorem 5.4.20 that (2) implies (1^) for some ß E [Ru{oo}. But (1^) for ß e [Ru{cxo} implies (*) by Theorem 5.4.17. Now, (Iß) for ß G IR\{0} implies that co is a factor state by Theorem 5.3.30 (3) and if (1^) is true for ß G {(X)}, then co is pure by Theorem 5.3.37. Thus, co is a factor state in both cases, and it follows from Example 4.3.34 and the asymptotic
{co'''^}
abelianness that lim inf|r,--/y|-^oo
CO(T,, (Ai)'-'T:f^(An))
=
o}(Ai)
-
-
o}(An)
'Vy
for all
Z+ and all AI G ^. Q, i.e., co is an extremal invariant trace, then ß invariant state by asymptotic abelianness, and thus If
G
(1/j)
holds for
=
MCO(AT:(B)) by
=
co
is
an
extremal
o}(A)o}(B)
Theorems 4.3.17 and 4.3.22. If
hence
co
is assumed to be
(2)
and
(1^),
But if 31 has
for
a
factor state,
some
ß
G
one
[Ru{oo},
derives are
n-poini clustering äs above, completely equivalent.
and
unique trace-state, then every extremal (T, ß)-KMS state co must be Q, then co is the j? ^ 0 this follows from Theorem 5.3.30. (If ß unique trace and is automatically a factor state.) The equivalence and (2) and (1^), for some j? G (R u {00}, follows once again. a
a
factor state. For
=
196
States in
Quantum Statistical
The last Statement of the 5.4.4 in
case
position A
in
Mechanics
corollary follows
A, Proposition 5.4.18 in
case
from Corollary 5.4.7 and Theorem C, and the remarks preceding this pro-
B.
case
slightly annoying
feature of the
stability requirement (2b) of Corollary required for CD^(A) (D(A). One may
5.4.21 is the small order in 1 behavior
avoid this
by assuming stability in norm of the limits lim^_,oo <^ ^^^^- To be precise, assume that T satisfies a uniform Z^-asymptotic abelian property
more
in the
sense
that
^^||[P,T,^-''{^)]|| is
an
L^-function for all ^,
5.4.10 and its
subsequent
P
P* G
=
2lo and l sufficiently small. Proposition
remark then
yf
lim
=
that the M011er
imply
morphisms
ti^T,
t^00
exist
and
strongly
are
yf^. Define states
co^ '
of ^.
*-automorphisms =
Furthermore,
^rrf
by
a;f(^)=co((7f)-^(^))-
cD(,f(A})
lim t^OG
and then
CDJ^
are
r^'^^-stationary
states. Now the
lim that
co
is
a
=^
i-KMS state for
follows. First, note that the states
condition
llcof -a)\\=0
;.^o
implies
stabihty
"
some
co^
value
have
a
jS
G [R u
{ib oc}.
This is
property of return
to
seen äs
equilibrium,
i.e.,
^_lim^<(T,(/()) Jm^<(Ti^T,(^)) =
=
<(yf (^))
=
o^A]
and
lim t-^
Thus, letting
T
-^
oo
and xS'
a}^(if(A))
=
co(A]
.
00
>
-
<(T,(^)-^)/I
oo
=
in the relations
-i
/'j^<([P,T,(^)]) f dtcD^_^([P,T,(A]]]
,
JQ
co^(A-,s(A})/l
=
-i
,
Js we
obtain
/oo J^<,([P,T,(^)])
(*)
.
-00
Theorem 4.3.17
implies
godic with respect
that the two states
to the action
rf
=
co^^
=
7^1^(7^)"^
coo
(y^)~^
are
centrally
er-
and hence it follows from
and
Stability Theorem 4.3.19 that the
a
co^^
are
either
equal
or
disjoint.
stability requirement, by 0;^^ straightforward extension of the argument used
for small /l
the
and
and
co^^
Equilibrium
But
\\co^^
co^H
197
<
2
cannot be
to prove
disjoint by Corollary 2.6.11. It
follows that
cüf for small A. Hence,
letting
l
-*
0 in
=
(*)
ca^:^
we
find the Standard
f 'dtco([P,r,(A)]) J
and
is
=
stability
condition
0
-C
value
ß G [Ru{ cxo} stability required Corollary 5.4.21 could be viewed äs a stability against contamination of the System. The perturbation of the Hamiltonian represents the introduction of an impurity into the System. One could alternatively envisage another kind of stability, namely, that the System co
a
r-KMS state at
(9l, T)
some
.
for the state in
The
in the state
co
is stable in coexistence with another System (91', T') in a in the sense of condition
CD', i.e., the joint System (91 0 91', T 0 T') is stable of Corollary 5.4.21. One then has directly (2) state
/dt(F(t}F'(t)-G(t)G'(t))=0 where
F,F', G,
and G'
are
defined
äs
before Observation l of Theorem 5.4.19.
If cü' has strong
clustering properties and is a i'-KMS state at value jß G !R, one can now proceed äs in the proof of Theorem 5.4.19 to show that o; is a i-KMS state at the same value ß, without assuming any purity of co. Assuming purity of co, it is enough that co' is not a ground, or ceiling, state to reach the same conclusion.
5.4.3. In the
Gauge Groups and the Chemical Potential
we described how the inverse temperature ß enters thermodynamic equilibrium from requirements of stability. But in the description of equilibrium states of the ideal Fermi and Böse gas in Sections 5.2.4 and 5.2.5, these states were also characterized by a second Parameter ju, the chemical potential. Equilibrium states cp were considered to be states which are Ty^-KMS states at value ß, where 1 1-^ y^^ is the group of of the is the i.e., algebra, gauge automorphisms t^-^y^ group of Bogohubov induced e^^H the \-^ the t on automorphisms by unitary group one-particle space. In Order to see how the chemical potential enters one has to examine the role of gauge invariance more closely. The setting of the problem is described by the following definition.
previous
subsection
the formalism of
Afield System is a sextuple (5, 91, G, 1,7,0-) where 5 is a identity, called the field algebra, G is a compact group, called
Definition 5.4.22.
C*-algebra
with
198
States in
Quantum Statistical Mechanics
the gauge group, and g e G\-^ jg is a continuous, *-automorphism group of g. Further ^
faithful, representation of G 5'^', the fixed-point algebra under the action of G, is called the observable algebra and /^ ^^ T/ is a con tinuous one-parameter group of *-automorphisms of 5^ called the time-translation group. Finally, ö" is a fixed *-automorphism of g such that into the
a^ The groups T, y, and
o are
oit
for alH e
[R,
interrelated
T/ö-
(jjg
,
=
{y,j]g
G}
.
by jgG
T^y^
,
=
y^T^
,
ö' e G.
Deüne the
S+ Then
-
ö-G
^i,
=
even
-
and odd
{^
5 is assumed
with respect to
e
subalgebras
S; ^(^)
-
to have the
A],
of
g_
g by
=
{^
G
S; (T(^)
following asymptotic
-
-A]
.
commutation property
T:
lim
||M,T,(5)]||=0
|/|->oo
if^GS,^Gg^,
and lim I/HOO
||{^,T,(^)}||-0
if^,5e5_. In typical applications the field algebra, or the algebra of quasi-local operations, is the algebra generated by creation and annihilation operators a*(/) and /(/), where the index / denotes the different particle types and their transformation properties under internal symmetries. The group of these symmetries constitutes the gauge group G. In an example of scalar particles the index / would ränge over 1,2, G f\l and G would be the n,77, for some dimensional torus T". An element 0^ G G is parametrized by n angles 0 ^ (Pi < 271 in this case, and the action of the corresponding automorphism ...
y^
is
explicitly given by
y,K(/))
=
e">"a^(f)
,
r,(/(/))
=
e-'"'.fl,(/)
.
As G represents inner symmetries of individual particles one expects y to commute with time translation T, and one also expects y-dependent quantities to be
macroscopically unobservable. Hence the name observable algebra for g^. The other concepts occurring in Definition 5.4.22 have been explained earlier (see, for example, Definition 2.6.3). Since T and y commute, it follows that ^ is globally r-invariant. Moreover, äs ö- G y(5 one has ^ C g^, and hence ^ is asymptotically abelian with respect to T. The results of the previous section then justify the KMS condition äs a criterion for a state co of ^ to be an equilibrium state. But in Sections 5.2.4 and 5.2.5 the equilibrium states (p of the ideal Fermi and Böse gases were defined to be KMS States at value ß for some group of automorphisms of the form ^
=
199
Stability and Equilibrium
(^^ is a one-parameter subgroup of G given by the potential. Conversely, t ^-^ ^t determines the chemical potential. But the restriction of ^ i-^ t/y,^^ to ^ is just rl^j, and hence o) (p\
IR
T^7^^ where
H-^
^
i>
chemical
=
=
This is achieved in two Steps: If
(1)
co
is
an
extremal r-invariant state of
invariant extension cp to related by
5,
9l=
for
and
is
has
an
extremal
T-
^^e
g ^ G.
some
co
co
yg
If, in the Situation described by (1),
(2)
^, then
and any two such extensions (pi and (^2?
a
T-KMS state at value
tinuous one-parameter KMS state at value ß.
subgroup
t
is
Uco
ß
\-^
a
faithful
1I^\{0},
e
representation
then there exists
^^ of G such
that (p is
at
of 91
a con-
\-^
'^tl^-
results, and subsequently make several remarks 0, etc. pertaining to variants, the case ^ Before the actual proof of these theorems, we characterize the extremal We first prove these two
invariant states of field Systems by düster properties. In the case that the i, these properties follow already System is asymptotically abelian, i.e., a from Theorem 4.3.17 and Example 4.3.5. For general a we use a method of =
to characterize states with trivial even
proof which is similar to that used algebra at infinity in Theorem 2.6.5.
Let (g, 91, G, T, 7,0-) be a field System, and cp a i-in0. The follomng conditions of 3f. Then cp o a (p^ i.e., (p\<^^ are equivalent: (1) q) is i-ergodic, (2) q) has the clustering property
Proposition
5.4.23.
variant state
=
^ Js/
lim T-S
->
00
l
dtcp(Ai,(B)]
Furthermore, in this Situation
l
:^
/ J^
where the limit exists in the sträng operator
'r ^
<X)
7^ / l
i3
^^
=
dtn^(it(A)] topology. Finally, =
BA;
B
J^
all A^ A' ^ 5, B' G l if ^'5 ^ ^ SA', B
for
.
/"^
lim
-
T-S^oo
T l o
cp(A)cp(B}
has
one
l
(p(^)l
=
>J
"^.^^g 5_
G
G, where the
^nd SA',
B
=
convergence is otherwise. l +
uniform
in g, and
200
States in
PROOF.
Quantum Statistical Mechanics
Let cp be
T-invariant state
a
plements T on ,^, and E^p from Example 4.3.5 that lim y
>
o
-
oo
l
/
-
is the
on g. If U(p is the unitary group which improjection onto the t/^^ -invariant vectors, it follows
dtn,p(i,(A))ü,p
lim
=
^Js
Ae^. assumption of Definition
The last
following
/
-
l
dt
U,p(t)n,p(A}Ü,,
^Js
E,pn,p(A)ü,p
=
for all
-
70^00
5.4.22 and the relation above
imply
that the
Hmits exist
f^
l
lim
-
T
T-S-^oo
=
/ dtTL,,(l,(A]]Tl,,(B]ü.,,
^ Js
A^Bn,p(B)
(
=
(^
l
hm
-
/ dt7i,p(i:t(A))Q,p
-
y
\T-s-^oo
ib j^
SA^Bn,p(B)E,pn,p(A)ü,p
and, by uniform boundedness, the limits
\nn
-i-
T-S-^cc
T-S
M(A)= exist in the strong M (A) is
topology. Now, T-invadant, hence
if ^ e
/
dtn,,(T,(A))
Js
then M (A) commutes with
5+.
7c,^(g)
and
'
M(^) for^
e
{n^pm^U^p(R)}'
eg+.
If A E
5-5 then
^* G
^.nd
5-?
we
have
f^
l
(ü,p,n,p(AYE,p7i,p(A)Q,p)=
lim
-/J^
-
T-S^oo /
=
=
and
hence, applying
this
(A
on
+
particular, (p(A)
=
0 for all ^ G
-(ü,p,n^(A)E,pn^(AYQ,p)
A*)/2
5_ (p
Let
Ucp((j) 7r,p(g), and
be the
and
G
We
o
(A -A*)/2i 0
=
and this a
unitary implementing i-invariant, hence
=
(p
G
.
implies
that
.
er on
,^.
Then
M(A)U(p((j)
{n^p(^)uU,p(R)y
g_. now
prove
(1)
=^
(2). Ergodicity .
,
,
it is also
M(A)U^(c,) foT A
dt (p (A'' i, (A))
t3
-^r dtcpMA}A^) -^lun^,^^l T-SJs
E,pn,p(A)^^ In
(*)
,
of (p is
{n^(^)uU^p(U)}'
=
equi equivalent U
to
commutes with
and
Stability by Theorem
4.3.17. If ^ E
5^,
M(A) If^ e
it follows from
(*)
that
(Q^, M(A)a^)^
=
(p(A)'f^
=
201
Equilibrium
.
g_(**) implies M(A)U^(a)
=
=
(Q^, M(A)U^(a)Q^)^ 9(^)11 (Q^, M(^)Q^)11
where the last relation follows from the first part of the
M(A) for ^ e
-
-
0
=
=
0
,
proposition.
Hence
(p(A)'^
5_ and by linearity ,
M(A) for all .4 e
(p(A)'^
=
g. Thus
lim
-i^ /
T S^00 l
iJ
Conversely, (2)
dtq>(A^,(B]]
=
(ü^,n^(A}M(B}ü^)
=
9(A)9(B}
Js
(1)
=^
is
immediate consequence of Theorems 4.3.22 and
an
4.3.17. It remains to establish that
r y
'r :>
>
cx)
^l l
o
dtcp(Ar,(A')y^(Br,(B')))
=
e,,,BV(Ay,(B))cp(A'y,,(ß')}
Js
A', B, B' e'S-^, uniformly for g\-^yg(B'] are continuous, it follows for A,
g
^
g e G. Since G is
compact and g
i-^
y^l^)
and
that
(p(Ai,(A')yg(Br,(B'}))
equicontinuous in g when t ranges over R. Hence, the uniformity of the convergence in g follows once we can show pointwise convergence. But this follows is
immediately
from the
two-point
düster
properties
of (p and the
asymptotic
com-
mutation property lim |?|->oo
together We
with the fact that
now
\MA')B-SA',BB^,(A')\\=()
yg('S)
5
Lei
invariant state
21.
on
It follows that
0}
(5, ^, G, T, y, er)
possesses
algebra g. Moreover,
two
consequence of
cry^
=
y^a.
an
g
be
afield System,
and
co an
extremal
extremal i-invariant extension (p to are related by
T-
thefield
such extensions cpi and cp2 92
some
äs a
State the first main theorem of this section.
Theorem 5.4.24.
for
for ^11 ^ e G,
=
92^
Jg
G.
We first prove the existence of 9 by a method analogous to that used to Proposition 2.3.24. This proposition implies that cü has an extension to a state of g and, by applying an invariant mean to this state composed with T, we deduce that the set ^^ of i-invariant extensions of co to states of g is nonempty. But ^"^ is
PROOF. prove
202
States in
clearly
Quantum Statistical Mechanics and weak* compact, and thus it contains
convex
Krein-Milman theorem. If cp A(pi + invariant states of g, one has
(\
=
cpj
(/)2|fli
=
yj
extremal in
=
the
by
co,
uniqueness of
variant extension (p of
co
sequel set
denote
we
the C
by C(X)
=
The
c;(G)
/(^s),
{pf")
=
transformations, fix
cp}
=
(p2
hence
(p since (p is
=
extremal i-in-
an
.
of continuous functions
-algebra
6 G
on
the
define
we
(^^(ff)/)W
i/ 6 G
:
C(G)
/(f7^'^)
=
Indeed Qj,(G) is selfadjoint by Proposition 5.4.23,
-i-
lim r-5-^00
for ^,5
,
G
C^J,(G).
^
=
(G)
-
G,p\G ^i
It follows
(p(^'>
since
cp^'"^'^
=
l' dtcp('^^'W\g)
C(G(p\G). for ^ 6
5,
and
multiplicative
(p(-^'(3)'?''''(ff)
=
j^
an
Ci
G/
:^
g, uniformly for g E G. equivalence relation
G
Define
/
i
^
G coincides with
on
PROOF.
But
=
the
are T-
and
A)(p2\yi
-
by
closure of
norm
in the continuous functions
for all
},(p^\,.^^-\- (l
=
It follows that cp^
{g ^G](poy^^
=
X, and for /G C(G) and ^
OBSERVATION 1.
since
extremal element cp l, where cp^ and (p2
G.
e
j
co.
cp up to gauge
(p(g}fm for
(p\,.^^
=
an
0 < l <
and define
G,p In the
CD
of
extremality
I)(p2,
E^^.
To prove
compact
-
~
on
G
by
^
0^1
^2
if, and only if, f(g\]
=
/(gi)
the Stone-Weierstrass theorem that
by
{/G G(G); ^1
^2 ^
^
f(gi]
f(g2)}
=
=
C(G/ ^)
.
because -
^2 ^
(p(yg, (A))
=
for alM
(p(yg,(A})
5
G
^(p(^)-(p(y^^^^,(^))forall^Gg ^
t/20'r^
^
<^^/^
^ ^2 e
G,^^i
.
Hence
CUG)-G(G.AG) OBSERVATION 2.
Let J^/,/ G, and identify C(Ki\G) with the
=
.
1,2, be closed subgroups of the compact
space
consisting
invariant under all left translations
of the functions in
C(G)
group
which
are
l(ki}^ki G ÄT/,/ 1,2. It follows that each -isomorphism of C(y^i\G) into C(K2\G} which commutes with the right translations P(9]i9 ^ GI is of the form X(h] for some /z G G such that K2 hK\h~^ =
=
.
PROOF.
The
continuously
transposed onto
the
map a* pure
of
states
a
maps the pure states
Ki\G
of
C(Ki\G)
and
K2\G
of
hence
C(K2\G) defines
a
and
Stability a*
homeomorphism
K2\G
:
which commutes with
Ki\G
^-^
Equilibrium translations
right
203
by
g eG.
Thus
cT*(^2^)
(T%K2eg)
=
(j\K2]g
=
=
(K^h-^]g
for all g e G, where /z G G is any element such that ff*(K2) obtain o ).(h]. Since af,feC(Ki\G}, has to
we
=
=
1(^2), ^2
^2,
^
we
eK2,f
argument with
(of](k2^g]
=
replaced a~^
now resume
the
h'^k^^
shows the
K2 We
^(k2)(af](g]
=
and hence
C(Ki\G),
e
a
.
under
invariant
all
must have
f(h~^k^^g) for all k2
K\h~^ By transposition
be
of the
proof
hKih~^
=
e
reverse
=
f(h-^g]
Kih'^ or K2 C hK^h'^ A similar inclusion, and .
.
uniqueness part
(P2 be two extremal T-invariant extensions of stabilizers in G. Now the element
af(g)
=
5,
to
co
of Theorem 5.4.24. Let cp^ and G
and Kj
f dgyg(A^i,(B)} JG
belongs
to 2l for all
^,ß
6
and thus
5,
the
gives
same
value of (pj and
that
(
=
jdgvf\g}
lim
=
T-s^oo
lim
=
T-s^oo (
l
^ ,^
-
\JG
Js
-i-/ dtcpJ\JQfdgy,,(A'r,(B)) l
Js
:^
(5)\
(^)
^(9V^^\ \\K.\G~] where
from
we
used
I^(KX\G)
Proposition onto
5.4.23. It follows that there exists
L~(K2\G)
defined
K-pf' for /4 G
'
a
linear
isometry
V
by =
^i^'
,
g. Since lim T-S^ooT
uniformly
in ^f G
L_ C^
l^'dtv(^-^'^^\g)
v(^^(g)
=
JS
G, it follows that
K(
=
for v4,5 G g. If r/ is the representation pointwise multiplication, one then has
r2(K/)
=
<,(-)^r)
of
/ 6 C(KI\G). As the f, are isometric, one C(A:i\G) onto C(A:2\G). The computation =
on
L~(Ki\G)
obtained from
Kr,(/)F*
for
{F(p(Ä)
C(Ki\G)
cp,(y(A])
concludes that Kis
=
{p(h)V(p^{'^}(g)
a
*-isomorphism
of
204
States in
Quantum Statistical Mechanics
shows that F commutes with the
yields ^g^G such that
q)^^~^
=
right regulär representation for all ^ e ^, i.e.,
p, and Observation 2
l(g)(p\'^^
^2(^)-^l(V(^)) for all A ^^. This ends the proof of Theorem 5.4.24.
The main result
on
Theorem 5.4.25. I- KMS
State
the existence of the chemical
(5, ^, G,
Let
over
^
at
T, 7,
valiie
ß
G
a) be afield [R\{0} such
potential is
the
following.
System, and CD an extrema! that the corresponding re
presentation of^ is faithfiiL Then each extremal i-invariant extension cp of CD
g is a KMS State at valiie ßfor a one-parameter groiip of ^-automorphisrns of g of the form t H-^ '^r?^,? where t \-^ ^^ is a continiioiis one-parameter siibgroup ofthe center ofthe stabilizer G^^ {^ G G; cp o y^ cp} ofcp. The groiip t \-^ ^f is uniquely determined by cp. to
=
PROOF.
=
The proof of this theorem
requires at several points the spectral theory of i.e., the theory of the unitary representations of G. Since we have not developed this theory for general compact groups, we will content ourselves with the group G,
proving
the theorem for the
slightly more transparent case of abelian groups G. The accomplished by replacing characters with irreducible representations (which are automatically finite dimensional), and products of characters with tensor products of representations. As a final remark on non-abelian G, note that if cp satisfies the ry^s-KMS condition at value ß G IR\{0} for some one-parameter group t^^ ^fin G, then c/? is ry^- -invariant by Proposition 5.3.3 and äs (p is T-invariant it follows that ^^ G G^p for all t. But if 0^ G G^^, then cp (p o y^^ and hence y^-^^tj^^yg T^ty^^, by Proposition 5.3.33, thus g~^ ^t9 C/j i-^., / ^> C^ is contained in the center of G. Throughout the sequel of the proof G is assumed to be extension to general G is
=
=
abelian. Let
{^^, Ti^/,, U(p,^(p}
be the covariant
representation ofthe G*-dynamical System G^/,, T x 7} denote the corresponding J^F* -System, i.e., 9JJ T^(p(^]" etc., and let Jy^"' and ^M^'^ be the fixed-point algebras ofthe action of G,p on g and 9JJ, respectively. By integrating over the compact group G,p one easily shows that Wf'^ is the weak operator closure of 5^'% but in fact we can show
{Jy,
[R
X
Gfp, T
X
y},
and let
{9}i,
(R
x
=
-,
OBSERVATION l
m^^^
PROOF.
Let G be the dual group of
W({l}] be the
S,/,
=
=
Ti,p(^S)"
G, and if
{Ae\^'^^y,(A]
corresponding spectral subspace by
of
.
7 G
G, let
=
y^(g]A}
^ (see Definition 3.2.37). Define
C G
E,^
=
[j^^G]
there is an^ G
5'({z})
with
q)(A) ^ 0}
.
the subset
Stability OBSERVATION 2.
If X,
PROOF.
then^T,(^')
G
^^
=
G^
^
-
z' ^ ^(p there are^ 5^(nO and äs
L
by Proposition 5.4.23, additive.
;((^)
l for all ^ G G J l for all 7 G
^'
S^''(/),
^
^r(^(^T,(^0)
SJ
with
.
(p(^) 7^ 0, q)(A'] +
^(^)^(^O 7^
-
,
0
0. But
,
0 for some t. Hence 77^ G S
(/)(^Tf(^')) 7^
it follows that other
it follows tbat
(^(A)
=
-
G^,
^ j^
the
On
i(g]
205
Equilibrium
of G with annihilator
subgroup
g^'(x),
G
-i^ /
lim
=:
closed
a
G^^={^e G; S^ {^ G G;
r-s^00
S^ is (p(yi*)
is
Z^
and
if v4 G
band, is
Z^^
sinee G is diserete. In
general
have
we
9(l,(A^^-W}9(A} if y4 G
5^'(/()-
Thus if ;^ G
and ö' G
S(p
G
must have
we
G.CS^ Conversely,
if g 6
2^,
and ^ e
^''(x)
Hence ^ e
G,p
and the
reverse
;^(^)
=
l, i.e.,
.
then
=
(p(^)
=
0
if;(ei;^
ifx^S^
=
.
incltision
S^CG, holds.
OBSERVATION
^eg'W.xeZ^,
3.
TO*^"
is
the
von
Netjmann
algebra generated by
the
i.e.,
XR^'"^'
= -
lx6"iU 5''(^)|" ^
'^
PROOF.
If ^ G
g^(;(),7
G
Z<^,
/
P
>
r..
Observation 2
implies
that A G
9}Z^^
and hence the
inclusion D is valid. To show the converse, note that any ^4 G 9J1 has the weakoperator convergent Fourier expansion
^-^^(X),
^(%)-
[ dgx(g)yg(A)
.
-^^ /ec
Thus if ^ G
^Jl^"^ ,
it follows from the
of this
uniqueness
(x(g)-^)A(x)
G^
=
if for all g G G,/, and all x ^ G. Hence ;( G servation 2, the reverse inclusion follows since
expansion
that
o
.4(7) ^
0 and
äs
G^
==
Z^ by
Ob
206
States in
Quantum Statistical Mechanics
A
^A(x)
=
.
/es,,,
We
now
finish the
proof of Observation
l, The inclusion
m^'^D7i^,(^df is evident since 5l that ifA G
exists
a
^
=
g^.
To show the converse, it
y (;;:), X ^ ^(p^ then A G G y (x) with (-/)(5) 1. =
suffices, by Observation 3, to show S,^, by Observation 2, there But by Proposition 5.4.23 the limit
n(p(^)"
But since / ^
-i^ l dtTi^,(i,(B]A)
lim
T-S^OQ l
ö
exists in the strong-operator follows.
cp(B)A=A
=
J
As
topology.
Ti,p(^)" Cn,p(^'^'^]" <^'m^'^
Since
.
i:t(B]A
G
5^
31 for all t, Observation l
=
it follows from Observation l that
7r,(Ul)"
=
7E,,(g^'0'
and hence
Corollary 5.3.4 implies that the restriction of c/? to g*^'-' is still a T-KMS ß. Hence, replacing ^l by ^'^'^ and G by G^^, it suffices to prove Theorem 5.4.25 in the case that G.^ G, i.e., we may assume that cp is gauge invariant from now on. (Note that a G G,^ by Proposition 5.4.23.) With this assumption we prove: State at value
=
OBSERVATION 4.
C,^(G)
The set
of functions of the form
cj^G^cp(Ayg(B]] where
^,5
PROOF.
G
5, ünearly Spans
[C^^(G)]
norm-dense
a
is closed under
,
of
subspace
C(G).
multiplication, because Proposition
5.4.23
imphes
that
^^j\t,p(A,,(A']y^^(BT,(B']]]=^,,^B9(^^^^
lim
T-S-^ocT
-
S
,
^
when
A^ A' B^ B' ^
G
57 uniformly
in g. Thus
g^cp(Ay,j(B]]cp(A'y,^(B']] is contained in
[C^^(G)],
when
A, A'
,
B, 5^
G
5.j_,
and
by linearity
this is still true for
general A,A',B,B'^ ^^ We next show that äs 71(0
(5)
is
[C^^(G)]
strongly dense
in
is closed under complex conjugation. First, note that 9JI, [C2(G)] is also the norm closed hnear span of the
functions
c/^'^);^^(i:!^7_,/5)a,) where A^ß
9JJ. But B has
ß
=
a
weak-operator convergent
^B(x)
,
B(x}=
and hence it is =
x(g}B.
enough If
BQ,p
=
l dgi(g]y,^(B]
to consider elements B which
0, then
Fourier
expansion
,
-^^
zeö
y^^(B]
,
f/>^'^'^)
=
0 and there is
eigenvectors for y, i.e., nothing more to prove. If
are
and
Stability
^ 0. But since BB* G 9J1*^ and (p\^j^G is a KMS state at value ß, it Corollary 5.3.9 that (p|s;j{G is faithful and hence (p(BB*} ^ 0, i.e.,
then BB*
BQ,p ^ 0,
follows from 0. But
5*n^ ^
y,j(B~^)
=
7.(gW,
and hence
,p(^^^\g] for
Q^(G)
suitable C G 9Ji Now,
a
lations
by
G
is
^(C.B^^(g)
=
globally
left and
=
r/'''(^)'^)
p(^)(p(^'^)
,
N^ normal
=
{g^ G-(p(Ay,^(B))
subgroup
But -
is
A',,
W)B
trivial
6
cK(5 by
-
if h e
because
W({x}).
.
in the
äs
proof of Observation
5}
C(G/N,)
=
and
N,,,
.
Be')X''({x})
for
some
then
G.
;/
But
i(h)Br(B
the definition of
'^^^^
of G and
[C2(G)]
B
invariant under trans-
for all ^,5 E
cp(AB)
=
cp^^^-
=
Hence, it follows from the Stone-Weierstrass theorem, l in Theorem 5.4.24, that
a
right
the relations
by
X(g)cp^^^^^
is
207
Equilibrium
N,/,
i(h}B))
-
and
=
v((B
-
/.(h)BY(B
-
-f,,(B)))
=
0
äs
(5-H^5r(5-^5)e9Jt^ and
(pljjjG
is faithful it follows that B
=
x(h)B
find that B
Fourier
=
y,,(ß)
.
yjj(B) for all B G 9JJ. But 7 is a faithful faithful representation of 5 äs we shall
expansion we By representation of G on 5^ ^.nd TT^^ is a subsequently show. It then follows from the last relation one
=
h
=
e.
Hence A^^,,
=
{e}
and
concludes that Observation 4 is valid.
To establish faithfulness of n^^
Fourier
expansion
of
n(p(A)
we assume
that /^ G
5
and
7l^p(A)
0. But then the
=
takes the form
7ü,X^)-^7r,,(^(7))
,
/G
where
^U)=
[ dgx(g)y,,(A) 65
.
JG
7i^(^(x)) 0 by the uniqueness of this expansion, 7r^(^(7)*^(x)) 0. Since^(7)*.4(x) G ^, it follows from the faithfulness ^(/)*^(z) 0. Thus A(i) 0 and y4 0. However
=
=
We
value
now
ß,
=^
complete
the
and of
hence
7r^|,)j
that
=
proof
of Theorem 5.4.25. Since
(p\,^i
is
a
i-KMS state
at
have
we
j dgcp(A^^,(A']i^,(y,j(B-^i,(B'm j dgq>(y^(B^i,(B']]A^i,(A']] =
when A, A'
,
B, and B'
this relation and
are
entire
Proposition
analytic
5.4.23 that
for
i.
If ^,
yl^
^, 5' G
5
it follows from
208
States in
^.
Quantum Statistical Mechanics
l dg (p(^*7^(T,-^(5*)))(p(^%((T.X50))
B
Now,
(p\c!^
äs a
consequence
l dg cp(y,^(B-^)A^)(p(y,^(B')A')
.
=
j^(/)(^*y,(T,yK5*)))9(^%(t//j(^0)) This relation extends =
T:
-.iß (B)*
by linearity
denotes the scalar
( ) Replacing B' by i_iß(B'] ,
are
entire
for
=
in
product
analytic
^^^(y,(^*M*)^(7,(^0^0 r-analytic A^ A'^ B^ B'
.
^. Using
äs
in this relation
(V-'/'W.^),,p(-^' S')) If AI and EI
-y
to all entire
this may be written
(^^(-.,-(S),^)_^('l',r,{S')) where
^B'.A
using S^; c g^, J5?. c5^, g^g_ cg_, g_j^^ c g_, and the fact 0, it follows that both sides of this equation are zero if SA>^B ^ SB', A, and
that
Tjß^B*}
=
=
(^(^5), 5^(5' ''))
L"(G]^ we
and
(sf](g]
f(g~^}
for all g
^
G.
obtain
(^(-4.fl)^^,^(T-,(S').^'))
and a/ G
T
,
=
C,/
=
l,
.
...,/7,
we
obtain from this
relation
(^a,V^-'^^(^'-)'^'),(^(^' ^')) ('^a.(p(^"^'),V'-'^(^')-^')) V/^i V/^1 =
/
As
[C^^(G)\
=
linear operator S
.
/
C(G] by Observation on /^~(G) by
4 it follows that
{"^a/c/)^"^"^'^:
we
may
^/,5/ entire analytic for
consistently
define
a
T
/-l
and
5(5;] (p(^"^'-)) ^;^a,.(/)(-'^(^')'^') a,
/
\/=l
.
/=!
One then obtains the relation
(S^,cp] for
l//, c/) D(S]. Hence Using the relations
iS is
l(^)./)(^^' ^) one
G
a
=
densely
r/)(>'^^^)' ^)
=
(ilj,S({)]
defined
,
Symmetrie operator,
p(^)(p(^' ^)
then deduces that S commutes with the left and
on
L^(G].
Since
51(^)
=
X(g)S
=
c/)(^^'>''^(^))
,
right regulär representations
of
for all g, it follows that 5 commutes with the
projection
/
=
/ dgx(g)).(g) JG/ dgx(g)p(g) =
JG
onto
the one-dimensional
Hence S is
subspace of L- (G) spanned by the essentially selfadjoint with spectral decomposition of
S=Y,S(X)E, XÖ
,
element g the form
^
x(^)-
Stability where
5'(%)
U.
G
for all ;( G G
a
Now,
(^^^'-^^
the set of
äs
pair A^B
of
spans
If
Equilibrium
209
Observation 4, there exists
such that
r-analytic elements
Ä^
C(G) by
and
.
defines
one
j dgW)yg(B}
B-,=
,
JG
then
is still
Bj^
since yi
t-analytic
=
Moreover,
ry.
(^(^'%)=^,(p(^'^)^0 and in
particular
^
v(B\Bj) Furthermore,
B
=
0
.
satisfies
T:iß/2(B*)
=
=
W}V^''' '\e)
.
We then compute
0 <
,p(5i5j)
^((T_,.;;/2(ß))(T_,^/2(5))*)
=
=
V^--*)(e)=5(x)
=
and this
implies
^(Z) It follows that ;S is
a
>
0
.
strictly positive selfadjoint operator
and hence
S^^
exists for all
[R.
t G
The weak
clustering
of q)
implies, by
5.(^(^,s)
particular using
B
=
5^,
/
G G.
argument which is by
5'(^(>',s))5'(^('<',fi'))
now
Standard, that
.
etc.,
5to') for all elements %,
=
an
But this
=
5(;f)5(x')
implies
5"((p(^'^)(/)(^'^'))
::.
5''((p(^'^))5''((^(^''^'))
and hence
t^S^' .5-^' defines on
^f
G G
M~S''
one-parameter group of *-isomorphisms of C(G) acting by multiplication
a
L^(G].
Since this action for each
follows from Observation 2 in the a
=
t commutes
with left and
proofof Theorem
such that Ad
~S''
=
l(^ß,)
.
right translations,
5.4.24 that for each
t
it
there exists
210
States in
Quantum Statistical Mechanics
But äs the automorphism Ad ?' of C(G) is uniquely determined by the underlying homeomorphism of G, ^^ is unique, and hence ^ x^^ is a continuous one-parameter i
group in G.
Thus, f^t
is
a
=
T,7,=,
continuous one-parameter group of *-automorphisms of g. G 5 äi's entire analytic for the two commuting groups
Now, if A^B
the one-parameter
er
and T, then
curve
^^^(^,-. in L- (G) has
analytic
an
extension to
this extension is
so
=
iß.
=
But
=5-"/^/.(^7,T,(5))
(p(^y^=,y^T,(5))
given by
5
t
=
v(By_,,(A))
=
(p(yg(B)A)
.
particular
(p(Aff^ß(B)) which is
just
In the
hence the
the KMS condition at value
course
of this
uniqueness
=
ß,
showed that TT^^ is a faithful representation of of the group t^^ C^ follows from Theorem 5.3.10.
proof
we
Jy
',
We end this section with several remarks a
concerning Theorem 5.4.25. If G is t^^ ^t is defined by an element called the chemical potentiell of cp. Since
compact Lie group, the one-parameter group
in the Lie
algebra of G which may be any two extremal t-invariant extensions cp^ and (^2 of
co are
related
transformation, Theorem 5.4.24, it follows from Proposition is uniquely determined by CD up to the transformation
by
a
5.3.33 that
gauge t^^
^f
(t^^,)^(t^g-^^,g) for g
G.
if
G for some (p, or G is abelian, then the chemical Hence, G(p Potential is uniquely determined by co alone. Note also that / 1-^ C^ is trivial, i.e., the chemical potential vanishes, whenever G(p is trivial, i.e., cp has no gauge T symmetry whatsoever. In the usual case G [R/Z, the chemical potential is a real number, and if it is nonzero cp is gauge invariant, i.e., T for G^p ~
=
=
=
all (p. In the Situation covered
subgroup [R X G by
of
G(p, i.e., t\-^^f
Theorem 5.4.25
by is
periodic,
Go-KxG,,, Then HQ is
HQ
=
that
assume
and define
^^
subgroups GQ
{(t,^,)',teU}
is
a
closed
and HQ of
.
closed, normal subgroup of GQ and GO///O is compact. Now an action of GQ on g, and since (p is extremal t-invariant, cp is extremal tQ-invariant. Assuming for simplicity that a i^ i.e., 5 is asymptoTO
=
T X
a
7 defines
tically abelian with respect to T, that g is asymptotically abelian
it follows
easily from
with respect to
the compactness of ^^ t^^ity^,^ and hence it follows
Stability
and
Equilibrium
211
from Theorem 5.3.32 that the
decomposition of cp into extremal KMS states ß, decomposition into extremal HQt\^ity^,, invariant states. (This decomposition also coincides with the central decom position by Theorem 5.3.30.) But then it follows from Theorem 4.3.37 that there exists a state (p e ^(E^^) such that this decomposition is given by for
at
value
coincides with the
(p(A)
=
l
dgq)(T:Q^g(A))
,
^GO/HO where
dg is normalized Haar measure on GO///O, and one may choose (p to be ^i-^T^y^^-KMS. Thus, all the translates ^OTO,^ are Ty^-KMS, and they are extensions of co to g, but they are not necessarily time-invariant states, they are only periodic in time. In our treatment of the chemical potential we assumed some form of asymptotic abelianess of the System (g, R, T). This assumption is usually not satisüed in quantum spin Systems, and for the treatment of such Systems it is more appropriate to consider co which are invariant under space translations in some direction in addition to being time invariant. If a is the space-translation automorphism, then it is natural to assume that l
for y4,^ G
g_t,
and then
lim cxo l
\\A(x,,(B)
-
A,BOCn(B)A\\
=
0
*
one can
build up
a
theory
of the chemical
considering extremal a-invariant extensions (p of co to mains valid in this setting by virtually the same proof.
potential by g. Theorem 5.4.25 re-
Finally we note that a modified version of Theorem 5.4.25 remains valid 0, i.e., when co is a faithful extremal T-invariant trace-state of ^. In jS this case an extremal i-invariant extension (p of co to 5 is not necessarily a trace-state. Nevertheless, there exists a continuous one-parameter subgroup t^-^^t of G(p such that cpisa. KMS state (at value -1) for t^-^y^^. This follows by essentially the same proof, normalizing such that Ad S^^ ^^^-t)when
=
=
5.4.4. Passive
Systems
Passive states of a äs
states
dynamical System (^, T) satisfying
were
-io}(U*ö(U)} for all U G
'^o(^)nZ)((5),
>
introduced in Definition 5.3.21
0
where ö is the generator of
T
and
^o(^)
is the
connected component of the identity of the group ^(91) of all unitary elements of 9l. This notion reflects a property of stabihty of equilibrium states
which is
basically
kinematic. For Gibbs states of finite Systems it is
a con-
sequence of the principle of minimum free energy. This was discussed in Section 5.3.1 prior to Theorem 5.3.15 and it follows from this discussion that
io}(U*ö(U))
is
exactly
state CDU where
the difference in energy of the state co and the perturbed co((7M(7). We next demonstrate that this energy
cot/(^)
=
States in
212
difference
Quantum Statistical Mechanics
be
directly interpreted äs the energy transferred from the System undergoes a series of external changes. But this transferred energy is the mechanical work performed by the System and thus one establishes that a State is passive if, and only if, it is unable to perform work in a cyclic process. This is the second law of thermodynamics. Combination of this result with the characterization of r-KMS states at positive ß given by Theorem 5.3.22 then provides a different justification for the i-KMS condition äs a characterization of equihbrium. It is first necessary to have a "time-dependent perturbation" analogue of Proposition 5.4.1. can
when it
Proposition denote
U\-^Pt
t ^
5.4.26.
the
(91, T)
Let
infinitesimal ^ ^ is
P^
=
a
be
a
generator
C*-
W-dynamical System
or
of
Furthermore,
T.
and let ö
assume
norm-continuoiis one-parameter family
that
of selfadjoint
elements, It
follows that there exists morphisms i^ of ^ satisfying
a
unique one-parameter family of
^(^)-Tf(^(^) and TQ
(y4)
=
A^ for all
A G ^. This
family
+
is
""-auto-
/[P^])
given by
Tf(^)=T,(^) +
Y^i" Jo(' dt, Jor dt2--- Jor""'rfrK(Pj,
,f>^, Moreover,
one
=
rfT,(/i)rf*
\vhere Y^ is the unique iterative solution
^-^ one
.
has
tf (^)
i.e.,
[[T,(P),T,{^)]]]
=
ofY^
iT^r,(P.)
,
=
^ and
,
has
rf-^
+
r^^2"- /""'j/.Tjpj..-Tjp,) Z^"/^^i JQ JQ Jo
.
~^i We will not
give the proof of this proposition äs it is essentially identical to Proposition 5.4.1. The principal difference introduced by the time dependence of the perturbation is that the group property of T^ and the cocycle property of F^ are no longer valid. Nevertheless, one can interpret the time development of states and observables in the same manner äs when the per turbation is time independent. We next consider perturbations t\-^Pt with the property P^ 0 for t < 0 or that of
=
t>T and
we
also
assume
that t\-^Pt is norm-differentiable. This type of
Stability and Equilibrium
perturbation
be
can
thought of äs changing
213
smooth, and temporary, alteration of the
a
external conditions. But
the external conditions transmits energy to the System, i.e., mechanical work is performed by the external forces. Ifco is the State of the System at / < 0, then at time / the state is given by co/ if^oj and =
the work
performed on the System can be calculated by summation of the incremental energy changes of co/ for all t G [0, T]. Explicitly one divides [0. T] into 7V intervals (tf, tf^i) vvhich are sufficiently small that co/ is essentially constant on each interval and defines the work by A^
rp^o,^ cü)^
Thus in the limit 7V
i>
oo
y" y"co,,('p/,-A,-,) /
^
this definition takes the
integral
form
L'(^>,rj-!f\=rj,^(^-ii Jo 7o \dt ) V V ^r An alternative way of
calculating
subtract the energy of the initial state
the work co
=
performed
on
the System is to final state
coo from the energy of the
0,T(A]^CD(Y^r^T(A]Y^T] and
analogy with
the finite System, discussed
that this energy difference shoiild be following resiilt. Lemma 5.4.27.
Let
(^21, T)
be
prior
given by
C*-
to
Theorem 5.3.15, indicates
-zco(rf (5(r^*)).
This
explains the
W^-dynamica! System and o) a state perturbation of T arfsing from a norm-diff'erentiahle faniily {Aj/^iR of selfadjoint elements Pt ^ D(Ö) such that Pt 0 for t <0, t>T, and dd(P,]/dt d(dPt/dt). Where ö fs the T. generator of Itfollows that over
a
or
*3I. Fiirthermore, let i^ denote the
=
^
L"(a^)=jjdtco(r^(^^^^ where PROOF.
F^
is the
unitary element relating
T^
=
-Ko(r^^(r^/))
and IT-
The derivation d is closed and it follows from
analysis of Section
3.2.2 that
ff
e
D(d), t^ö(Y^^]
Proposition
Therefore
But the differential
=
equation
-ifjdtco(^^s(r^') rf5(^)) +
for
5.4.26 and the
is differentiable, and
^-^'>-^^-/oxr;<5(r-))
.
F^ immediately yields
States in
214
Quantum Statistical Mechanics
-/co(r?.5(r?*))
=
r dtw(r^,,{p,)S(r^*)
-
rf5(T,(p,)rr))
JQ
=
/^^^co(rfT,(^(p,))rf*)
-
.
Jo The second step
that r^ relates
T
the derivation property of ö and öi one can reexpress this äs
uses
and T^
-/(r^.5(r^*))
=
=
TÖ. But
remembering
/'rf?a;(Tf(<5(P,)))
-
Jo On the other hand
one
may
L''(co)
=
/
This establishes the desired
Theorem 5.4.28. over
\
"^ /
/
=
Pj^
io
find
r^
dtw(,^(5(P,)))
5.4.27, Theorem 5.3.22, and
the
following
Let
(^, T)
co z^ a
(i:^ß]-KMS
The work
.
an
approximation
characterization of i-KMS states.
be
a
state
performed
. for
every
on
C*-
=
for
or
W-dynamical System
some
ß
G
the System is
[0, oo]
and
co a s täte
and
co
is i-invariant.
positive, i.e.,
f. ,<,(<()),
0.
norm-differentiable family {Pt}t^u of selfadjoint perturba0 for t < Q and t > T.
tions P t e "il such that P t
is
Q
^l. Consider the following conditions.
(1) (2)
äs
=
equality.
Combination of Lemma
gives
///r^\
PQ
use
dtw((-^](P,)]=^0 vv
-
Jo
argument then
and
integrate by parts
rT
=
Itfollows that (l) =^ (2) and if there exists a group G and an *-automorphisms o/ 21 such that co is oc-invariant, a commutes \veakly OL-clustering, then (1) <^ (2).
PROOF.
(1)
^
(2). First,
assume
5.4.27. It follows from the
that
ÖL
and
i,
co
satisfies the condition of Lemma
{Pt}t^m
passivity property
of G
action
with
established in Theorem 5.3.22, and
Lemma 5.4.27 that
1^(0;) Second, if P^ ^ ^(<^), then
P'l In the C*-case
one
has
2.5.22 and it follows
-
one
=
-zco(r^^(r^*)) replaces
it
by
0
>
the
regularization
J- l dse-'^^\,(P,] V T^J
||P'/ -P/||
0 and
->
easily that L^'\(D)
L^(co)
-^
lim
=
/;
^
oo
\\dP'Jdt
L^(co}.
L^"(a))
.
.
-
dPt/dt\\
Hence
>0
^
0
by Proposition
Stability by
the argument of the
delicate. As
co
215
Equilibrium
In the W*-casQ the argument is
previous paragraph.
is i-invariant the group in the representation (g^^, Tifo,^)- But if P" more
and
is
implemented by a unitary are the above regularizations
T
little
a
group U(,j
of P/,
one
then has
lim n
>
||7i,XP;'-Pr)A||
/^5^--||7r,,(T,/^(P,)-A)^|| 7ü-'/2 /"rf5e--{||/',|| ||(i7(-V%^)-1)AII J Ti-i/^
lim
<
co
n
^
J
oo
lim
<
"^00
^\\(U^(s/V^}-^)n,,(PM} for all
i/^
=
^
and
similarly for dP^ /dt. It follows immediately from the uniforai strongly convergent sequences, and the Lebesgue-dominated con0 in the strong-operator topology. Theretheorem, that 7Cfy(rf" Pf )
G
,j,
boundedness of vergence
-^
-
fore, \
/
//yp"\
(wp/zx ^jj (^n,7r<(rr>W'ta,(^^jc/,(-0't<(rf")*a, <^)) =
and another
application
of the
Lebesgue theorem gives
L^(co)-
L^"(co)
lim
>0
//>oo
äs
before.
(2)
(1)
=^
We argue that condition
:
(2) implies that
co
is
passive
and then
apply
Theorem 5.3.22. Let U
belong to the connected component ^/o(^) of the unit element in the group unitary elements of ^. It follows that there exists a finite sequence {Aj}^ <^.
A^
^
n ^'''
^
7=1
(This follows because \\U\
-
U2\\
let d denote the generator of
Therefore, U [R such that
t
=
<
2
implies U\
and
T
assume
that
^2^'^ by spectral analysis.) 1,2, Aj G D(d) for all 7
Next
=
...
,A^.
D(d] by Lemma 3.2.31. Now if/is a twice-differentiable function on 0 for ^ < 0 and f(t) l for /^ > T we define U^ by ^7, 11 for f(t] G
=
U for ^ >
:=
=
r, and
f;^^g//i, ...g^^yg/7(^-7T//VK-^, for
yT/A^
< ^ <
(/'
+
l)r/A^.
Further define
.
i.e., P; and
Ut
are
linked
by
=
Ut
=
rf
and in
5.4.27 and hence
the dififerential
particular
Pt
by
--,(;^
'^ Thus,
perturbations
U
=
=
equation
iU,.,(P.)
Y^.
But P/ satisfies the conditions of Lemma
States in
216
Quantum Statistical Mechanics
-ico(Ud(W]]=L^(o3] the
Finally,
case
that U G
D(d)
but
>0
.
is handled
Aj 0 D(d]
by approximation.
We omit
the details.
Remark. There is
one
simple
but
L^(o))
interesting application =
of Lemma 5.4.27. Assume that ^ where co/ is
aged
äs
{Pt}teu^ Systems.
a
two
of the
identity
-zco(r^^(rf ))
=
5Ii
^2,
(g)
(T, ^^)-KMS state with ßj > 0, independent Systems. Next
z
T
=
TI 0 12, and
=
l,
2. Thus
consider
an
C^K, T)
=
co\ ^ 0)2,
can
be envis-
co
external interaction
of the type considered in Lemma 5.4.27 which temporarily links the As T is a product, ö ö\ <^i^ i^ Ö2, where (5/ is the generator of T/ =
and hence
L^(a))=Zf(co)+I^(co)
,
where
If (cü)
=
-m(r'^(ö,
^)(^?*))
Zf (co)
,
=
-rco(rf (z
,52)(r?*))
.
These latter
quantities represent the mechanical work performed on the first Subsystems, respectively. But co is a (f, 1)-KMS state where ^\,ß^t^^2, ß,t and hence
and second
^t
=
^ilf H by
+
j52Lf (co)
Theorem 5.4.28. Thus
=
CD(r^T(ß^öi ^^^ß2^^ ^2)(r^*))
setting ß^
(T,
-
=
l/T/
T2}L^,(cD)
one
0
finds
ril^(co)
<
>
.
Now if TI
could
^ T2, then co cannot be a (T, ^)-KMS state for any ^ > 0. Thus, it happen that L^(co) < 0. If, however, T\ > T2 then one must also have
/.{'(co)
<
0 and
-L^(co) -^f (co)
TI
-
72
^ -
n
This is Carnot's
states that the
limited
and T2
and
formiila which by (T\ T2}/T\ where T\ heat sink, respectively.
are
efficiency
of
a
heat motor is
the temperatures of the heat
source
Notes and Remarks
Sections 5.2.1, 5.2.2, and 5.2.3 The
study
attempts
of the CCRs and CARs
to combine
The CCRs radiation
were
theory
over
^
=
L^(R^)
was a
direct outcome of
quantum mechanics and classical field theory. introduced by Dirac [Dir 1] in 1927 in the context of
and
are
generahzations of Heisenberg's commutation rela-
tions in quantum mechanics. The CARs were introduced by Jordan and Wigner [Jor 1] in 1928 for the purpose of quantizing the electron field. These authors also proved that the CARs over an -dimensional Hubert space generate the füll 2"
x 2" complex matrix algebra. Since the late 1920s the theory of creation and annihilation operators has been developed by many people in a multitude of specific contexts, e.g., many-body physics, quantum optics, statistical mechanics, relativistic field theory, etc. Much of the analysis is, how-
ever, of
heuristic nature.
Rigorous mathematical analysis of the commutation relations, algebras, only began in the 1950s, e.g., essential selfadjointness of ^(/) was first proved in 1953 by Cook [Coo 1] who established that the (/) zH have dense ranges. Cook also gave a precise dethe of Fock scription space, introduced by Fock in 1932 [Foc 1], and the corresponding representation of the CCRs. Consequently, this representation is often called the Fock-Cook representation. a
and the associated
The abstract structure of the creation and annihilation operators was also notably by Friederichs [[Fri 1]] and Segal (see [[Seg 1]] for
studied in the 1950s,
description of the subject during this period). In particular Segal emphasized C*-algebraic structure and introduced several new features and results. For example, it was traditional to analyze the CARs over a real Hubert space I) in a
the
terms of "fields"
0(/)=2-^/2(a(/)+a*(/)) and their
"conjugate
fields"
n(/) but
Segal emphasized
the
=
-/2-'/2((/) _'(/))
advantages of considering complex l) and
a
single
algebraic uniqueness of the CCR algebra, Theorem 5.2.8, was by Slawny [Sla 1] by the method we have described.
estab-
field.
The
lished in 1971
218
States in
Quantum Statistical Mechanics
The Stone-von Neumann
uniqueness theorem, Corollary 5.2.15, was anby published proof was given by von Neumann [Neu 4] a year later. The lack of uniqueness for Systems over infinitedimensional spaces 1} caused a lot of confusion which was not fully clarified until the late 1950s, and early 1960s. Until 1950 it appeared that most people tacitly believed that a Weyl System over a general Hubert space I) had a unique regulär irreducible representation. Counterexamples appeared in various disguises in the 1950s. Segal realized that various canonical transformations of the Bogoliubov type were not unitarily implementable (see [[Seg 1]]), van Hove subsequently showed [Ho v 1] that the uniqueness hypothesis led to a paradoxical "orthogonaHty of two complete orthogonal sets of eigenfunctions in Fock space," and Haag [Haag 4] estabhshed a general result which imphed the existence of inequivalent representations. The work of Haag also clearly demonstrated that representations which are inequivalent to the Fock re presentation are crucial from the physical point of view. Roughly stated, Haag showed that each dynamics has a distinct set of relevant representations and different dynamics lead to inequivalent representations. (A precise Statement of this type is given in Corollary 5.3.41.) Another aspect of inequivalent re presentation was provided by the work, in 1954, of Gärding and Wightman [Gär 1]. These authors proposed a classification of the (regulär) representations of the CCR and CAR algebras based upon properties of a number operator and, in particular, a result of the type given by Theorem 5.2.14. Various versions of this theorem were subsequently proved by several authors, e.g., Deir Antonio, Doplicher, and Ruelle [Ant 1], Dell'Antonio and Doplicher [Ant 2], Chaiken [Cha 1], [Cha 2], and Courbage, Miracle, and Robinson [Cou 1]. Our discussion is an extension of this last reference. It should be emphasized that the CCR version of Theorem 5.2.14 may be false if /2fü(^) is defined by summing \\a(o(f)^\\^ over all /in a fixed orthonormal basis of t) (see [Cha 1] for Counterexamples) but the CAR version remains valid. In 1964 Haag and Kastler [Haag 9] proposed a C*-algebraic reformulation of quantum field theory which emphasized the quasi-local structure of the field algebra. They also clarified the importance of the global C*-structure and the existence of inequivalent representations for the discussion of topics such äs physical equivalence and superselection rules. Truncated functions were introduced in quantum field theory by Haag [Haag 5] who realized their convenience for describing düster properties (see Example 5.2.19). Analogues of these functions occur in other disciphnes, e.g., cumulants in probability theory, Ursell-Mayer correlation functions in classical nounced
Stone in 1930 but the first
Statistical mechanics, etc.
Bogoliubov exploited automorphisms arising from transformations of the one-particle space I) in his 1947 investigation of the equilibrium states of the Böse gas [Bog 1] and subsequently these automorphisms have been called Bogoliubov transformations. The names one-particle automorphism, or quasifree automorphism, also occur. Combination of the Fock state with a Bogo liubov transformation immediately yields a quasi-free state and this was one of
Notes and Remarks
the
of such states.
origins
They also naturally
occur äs
equilibrium
states of the
ideal gases and other noninteracting Systems. The general notion of free state äs a state whose higher-order truncated functions vanish
introduced
Robinson
219
a
quasi-
was
in-
and Shale and
dependently by [Rob 3] Stinespring [Sha 1]. Robinson concentrated, however, on translationally invariant states while Shale and Stinespring only examined gauge-invariant states. The general theory was subsequently developed by numerous authors, e.g., Araki, Balslev, Deir Antonio, Manuceau, Powers, Stornier, Verbeure (for a review up to 1969 see the proceedings of the Cargese Summer School [[Car 1]]). One of the most striking early results was due to Powers who constructed a one-parameter family of mutually nonisomorphic type III factors äs the weak closures of representations of the CAR algebra from quasi-free states [Pow 1]. Explicitly, the weak closure of 7r(^;(^), where co;^ is the gauge-invariant quasi-free state given by
cü,(a(/ra(0)) is
factor of type Subsequently, the
=
%,/)
,
in the sense of Definition 2.7.24, for 0 < A < 1/2. quasi-free factor states have been completely typed by properties of their two-point functions and criteria for quasi-equivalence of such states have also been given. For example, every gauge-invariant quasi-free state of the CAR algebra is a factor state and if co^, and coA2 are such states with two-point functions given by a
ni;^/i_;
coAMfT^(9}} then
7ico,,| is quasi-equivalent
(l are was
to
=
(g.Aif}.
Tico^,^ if, and
^'-1,2,
only if,
A\'~Aj
and
-.4i)^/^-(l -^2)^^^
both Hubert-Schmidt operators. The sufficiency of these latter conditions derived by Dell'Antonio [Ant 3] and Rideau [Rid 1] while the necessity was
established
by
Powers and Stormer
[Pow 5]. particular n^o^^ is quasi-equivalent to the Fock representation if, and only if, A^^~ and (H A}^'~ H are both Hubert-Schmidt operators. But äs /l > ((l /l)^/^ 1)^ the latter condition is redundant, i.e., Ti^y^ is quasiequivalent to the Fock representation if, and only if, A is of trace-class. But the trace of A defines the number of particles in co^ Note that in
-
,
Tr(^)
=
^cü^(fl*(/,.M/;-))=K)
.
/>!
Hence,
CDA is normal with
respect to the Fock representation if, and only if, particle number. It is interesting to note that the equivalence between A being trace-class and Tift;^, being quasi-equivalent to the Fock representation 71^0^ is formally clear from the identity
it has finite
coA(B}=Tr(r(A)B)
,
220
States in
Quantum Statistical Mechanics
where B is
represented in the Fock representation. One uses that T(A] is of if, and only if, A is of trace-class. The generalization of these results to the CCR algebra and nongauge-invariant states was obtained by Araki [Ära 10], [Ära 1l], Araki and Shiraishi [Ära 12], and van Daele [Dae 3]. trace-class
Sections 5.2.4 and 5.2.5 The ideal gases chanics (see, for
are
discussed in most Standard textbooks
on
Statistical
me-
example, Huang [[Hua 1]] or Landau and Lifschitz [[Lan 1]]) and it is common to perform the thermodynamic limit implicitly. Rigorous proofs of the existence of these limits are in fact quite recent. The most difficult case, the Böse gas in the region of condensation, was treated by Cannon [Can 1], and Lewis and Pule [Lew 1], in 1973-1974. But both these works were partially based upon unpubhshed lecture notes of M. Kac. Our discussion is somewhat different and is based upon unpubhshed lectures of D.W. Robinson. The algebraic structure of the Gibbs thermodynamic limit states had, however, already been analyzed by Araki and Woods [Ära 8] for the Böse gas and by Araki and Wyss [Ära 9] for the Fermi gas. The Araki-Woods paper showed for the first time that equilibrium states lead to type III factors. The ideal Böse gas exhibits more complex condensation phenomena than we have described. Theorem 5.2.30 estabhshes that if A/^ is a parallelepiped with edges whose lengths tend to infinity in a homogeneous manner then at high densities a finite proportion of the particles occupy the lowest energy state, i.e., the ground state. But it is also possible to have a finite proportion of the particles occupying the energy levels in an arbitrary small band above the ground state with no macroscopic occupation of the ground state itself. This latter phenomena is referred to äs generalized condensation. In general there are two critical densities p^^ < PC- If the density p < Pg^ there is no form of condensation, if p e (p^^, p^] generalized condensation occurs and for all p > p^ the System condenses into the ground state. For example [Ber 2], if AI is a parallelepiped with sides LI > 1/2 > > ^v with L/ oc and (L2 -^v) / ß then p^ cx) but log Z2/(l3 Bn~^ Zi Thus of -^ .Z/v) properties Pg^ condensation are sensitive both to shape and to boundary effects. The general theory of both forms of condensation was developed by van den Berg, Lewis and Pule [Ber 3] and a variety of examples given by van den Berg, Lewis and Lunn [Ber 4]. But these papers were the sequel to various earlier investigations, see, for example, Girardeau [Gir 1], Schultz [Seh 1] and van den Berg and Lewis [Ber 1]. Other references and background material are given in these -^
>
.
-^
.
.
references. The allusion to
negative temperatures made in the footnote following by reference to the models of magnetic materials discussed in Chapter 6. The equilibrium states for these spin Systems are defined for all values of ß and a change of sign of ß corresponds to a change Lemma 5.2.25
can
be understood
Notes and Remarks
of
sign
of the interaction. Thus if the temperature could be lowered
221
beyond
absolute zero, an attractive interaction would immediately become repulsive, and vice versa. Consequently, ferromagnetic materials would become anti-
ferromagnetic,
and
conversely.
But these two
phenomena
are
physically quite
distinct and hence the passage to negative temperatures would be heralded by a sudden change of magnetic properties. Such transitions have been observed, for
example by Abragam
fluoride and the
same
and Proctor
authors
give
[Abr 1]
a more
in
experiments
on
lithium
detailed theoretical discussion in
[Abr 2]. Section 5.3.1 The KMS condition Kubo
[Kub 1]
was
first introduced in quantum statistical mechanics by Schwinger [Märt 1] in 1959 äs a con
in 1957 and Martin and
dition satisfied
by thermodynamic Green's functions. This condition was systematically exploited in the analysis of Green's functions by Kadanoff and Baym [[Kada 1]]. It was first formulated in C*-algebra language by Haag, Hugenholtz, and Winnink [Haag 6]. The original formulation of this condition is the one used in Proposition 5.3.12. During the same period, 1966-1967, that Haag, Hugenholtz, and Winnink, investigated the algebraic formulation of the KMS condition, Tomita developed the essential points of the theory of the modular automorphism group (see Chapter 2). Although i t was apparent that some connection existed between these two formalisms no complete synthesis was provided until 1970 when Takesaki [Tak 3] published his investigation and elaboration of Tomita's theory. In particular, Takesaki proved that the modular automorphism group satisfies the T-KMS condition with ß l and that this condition uniquely determines the group T when the state is given (Theorem 5.3.10). In the intervening period, 1967-1970, various other authors working on the KMS structure established partial results in this direction. For example, Winnink [Win 1] obtained the uniqueness oft from the KMS condition and also [Win 2] proved that this condition implies invariance of the state and pointwise invariance of the center of the corresponding representation. Both these in variance results were obtained independently by Hugenholtz and the state invariance was also obtained by F. Rocca and M. Sirugue. The multidimensional version of the edge of the wedge theorem, alluded to prior to Proposition 5.3.6, is discussed in [[Stre 1]], while more general forms of the Paley-Wiener theorem, Proposition 5.3.11, can be found in [[H0r 1]], [[Ree 2]], [[Rud 1]]. The three-line theorem can be found in the books by Titchmarsh and Rudin [[Tit 1]], Proposition 5.3.5 or [[Rud 1]], Theorem 12.8. The earliest correlation inequality of the type encountered in Theorems 5.3.15 and 5.3.17 was given by Bogoliubov [Bog 2] who proved in 1962 that the Gibbs state o) of a finite System at inverse temperature ß satisfies =
\o)([A,B])\^
<
-
-ißo}(AA^ +A''A)co([B\d(B}]}/2
.
222
States in
Quantum Statistical Mechanics
This
inequality was extended to general (T,jS)-KMS states in 1972 by Wong [Gar 1]. Basically, it follows from the Cauchy-Schwarz inequality applied to the Duhamel two-point function, introduced before Theorem 5.3.17. A detailed derivation and an application is given in Chapter 6 (see Section 6.2.6, Lemma 6.2.51). Alternative upper bounds for the Duhamel function which improve the Bogoliubov inequality were subsequently derived for Gibbs states by Berman, Bruch, and Fortune [Ber 1] and by Roepstorff [Roe 1]. Falk and Bruch [Fal 1] and Roepstorff [Roe 1] also derived lower bounds. RoepstorfFs results were extended to general KMS states by Naudts and Verbeure [Nau 1]. This generalization relied partially on earlier work of Naudts, Verbeure, and Weder [Nau 2]. Subsequently, Fannes and Verbeure [Fan 1], [Fan 2] established a converse to these results by showing that the KMS condition could be derived from the Roepstorff inequality. Independently Sewell, partially in collaboration with Araki, had been attempting to prove the equivalence of a maximum entropy principle and the KMS condition for spin Systems (for details and references to this development see Chapter 6) and the Fannes-Verbeure result provided the strategy for pas sing from the entropy principle to the KMS condition. Sewell [Sew 1] deduced the correlation inequality of Theorem 5.3.15 from the argument outlined prior to this theorem and also showed that this inequality implied the KMS con dition. The converse implication followed for spin Systems by the earlier work of Araki and Sewell [Ära 22]. For general KMS states it was given in the lecture notes of Araki [Ära 13]. Generators of positivity-preserving semigroups on C*-algebras have been extensively studied in connection with the theory of irreversible quantum processes, [[Dav 1]], [[Eva 1]]. If the semigroup is norm continuous, one can give several equivalent characterizations of such generators, and the most complete result in this direction was proved by Evans and Hanche-Olsen in 1977, [Eva 1]. Garrison and
Theorem.
Let y be
a
y(^*) =y(A]forallA
boiinded map G ^.
The following conditions
(1) (2) (3) (4) (5) (6)
e^^ is
a
(h
y]~^is
IfA IfA
G G
-f
C*-algebra
^ with
identity
such that
equivalent'.
positive map for all ^ G [R+. a positive map for all large positive reals X. 0. ^+,5 G 51 satisfy AB 0, then B"'y(A)B ^+ and CD G E^^i satisfy CD(A) 0, then o}(y(A)} > 0. =
-
=
y(A^) -i-Ay(^}A y(1])
are
on a
>
^*y(1])[/
Note that in the
special
>
y(A)A -^Ay(A) for all selfadjoint A G ^21. y(^*)^ -f t/*];(t/) for all unitaries ^ G
case
y(1])
-=
0 conditions
(5)
and
condition
y(A''A]
>
y(v4*)^
+
A^y(A)
(6) follow
^. from the
Notes and Remarks
mentioned
prior
to Theorem 5.3.15 and in
latter condition is
1], is given by
^
where A^ is the
Proposition 3.2.22,
223
but in fact the
strictly stronger than (5) and (6). An explicit example, [Eva ^(C^), the complex 2x2 matrices, and y(A) A^ A,
=
matrix of A,
transposed
^
1
\ ^
/
b\
a
/
a
c
d)~\b
d
_
/
\
/
In fact the groups generated by a generator 7 satisfying the strong condition characterized by the generalized Schwarz inequality, i.e.,
Corollary [Eva 1].
Lei y be
y(^*) =y(AYforallA
a
linear map
on
a
C*-algebra
are
^ such that
G 91.
The following conditions
are
equivalent:
e^^(A^A) > e^^(A^y^(A)for allA^^.t^ tR^y(A''A] > y(A^}A + A*y(A} for allAe^.
(1) (2) One has
an even
better
of norm-continuous
understanding
completely positive maps on von pletely positive map between two Stinespring in 1955 [Sti 1]. Let ^
Neumann
algebras.
C* -algebras ^ and S =
^ (g)
^(C")
matrices with entries in ^. A linear map (/) the induced map (^ : %j \-^ S defined by
v((Aij]lj^,)
=
:
^
be the
H->
was
of
semigroups
The notion of
a
com-
introduced
C*-algebra of all
35 is said to be
by x
n-positive if
(
l, 2, 3, positive, cp is said to be completely positive if it is -positive for n For example, an automorphism of a C*-algebra is completely positive, while an anti-automorphism of a non-abehan C*-algebra is positive, but not 2-positive. For all n there exist maps which are -positive but not n + l -positive. There does not seem to be any deep inherent reason for studying completely positive maps among all positive maps except that they are relatively easy to handle due to the following result, which is often referred to äs Stinespring's
is
....
theorem.
Proposition [Sti 1]. If^isa C^-algebra with identity, and cp : linear map, then (p ts completely positive if and only if it has
(p(A) for
some
map F
:
representation t>
5t.
parable. If^isa to
//
n
of
^ and
von
^
=
^
i-^^()
is
a
the form
V^n(A)V
Hilbert space 5^, andsome bounded linear separable, then 5^ can be taken to be se-
on a are
Neumann
algebra
and (p is normal, then
n can
be taken
be normal.
The observant reader may have noticed that this proposition was actually during the proof of the generalized Schwarz inequality, Proposition
established
3.2.4, where it
was
also established that
a
positive, identity-preserving
map
224
States in
from
Quantum Statistical Mechanics
abelian
C*-algebra into another C*-algebra is completely positive. One a positive, identity-preserving map from an arbitrary C*algebra into an abelian C*-algebra is completely positive, in particular states are completely positive [St0 1]. During the course of the proof of Proposition 3.2.4 we established that the generalized Schwarz inequality
can
an
also show that
cp(^}(p(A^A)>(p(AYcp(Ä) is
always valid for a completely positive map cp. Araki proved in 1973 that this inequality follows from 2-positivity alone [Ära 23]. In 1976 Evans [Eva 2] proved that if c/?: ^ =^() is a /z + l -positive map, where n > l, then the matrix inequality ^^
(
(p(B'^B)~^).
In
>
((^;S)
B, A\, ^2,---, A,^ G^ (with the proper interpretation of
particular
\\cp(B-^B}\\(cp(A^A,}}l^^, and, setting
5
=
n
=
,
l this is
that if
versely
(cp(A^B)cp(B^Aj))l^,
H
M((p(A*Aj))lj^, For
>
just
the
generalized
((p(A*)(p(Aj))l^,
>
Schwarz
inequality.
.
One
can
bounded linear map cp: ^i> =^() satisfies these for all /7, then (p is completely positive. If ^ is a von Neumann algebra on a Hubert (or ^ space a
=
follows from (^
^
:
1-^
^ is
Stinespring's theorem and Theorem 2.4.26 completely positive if and only if it has the
C/)(^)
=
that
a
show
con-
inequahties
=^^())
it
normal map
form
^F:/(F, a
for
a
suitable set
erator
topology.
finite if The
{V^}
The set
C
=^(),
can
where the convergence is in the weak-opbe taken to be countable if is separable, and
is finite dimensional.
following
theorem is
a generalization of the derivation theorem, Corollary 3.2.47, generators of norm-continuous semigroups of completely positive maps. It was proved in some special cases by Lindblad and independently by Gorini, Kossakowski, and Sudarshan in 1976, and was extended by various authors until the final version was proved by Christensen and Evans in 1978 [Chr 1]. The latter paper contains a complete history and bibliography of the subject, and the final result is
to
Theorem
[Chr 1]. Suppose that t^ U^i-^e^''' is a norm-continuous semiof completely positive maps on a C"" -algebra ^ acting on a Hubert space . Then there exists a completely positive map (p from 51 into its weak closure ^ and a selfadjoint element H H* E ^ such that the generator y has the form group
=
Notes and Remarks
y(A) Here R
=
=
[iH,A]-{R,A}/2
In particular, if the maps e^'' normal and y has the form
y(A)
[iH,A]
=
.
denotes the anti-commutator
y(^)and {, }
(p(^}
cp(A)
+
are
normal the map cp
{R, A}/2
-
+
225
can
.
be taken to be
^ V^AV, a
for suitable F
J^().
e
theory for generators of norm-continuous, normal, completely positive semigroups on von Neumann algebras can thus be summed up äs follows: The
Corollary. Suppose that ^ G [R+ i-^ e^"^^ is a norm-continuous C^-semigroup 1 for Neumann algebra ^R on a Hubert space , such that 6^''(1]) te IR+. The follomng Statements are equivalent:
a von
(1) (2)
=
e'^' is
a
completely positive
(700(-^*^) and each
> =
n
map
for
each
^ G
on
all
[R+.
(y^i](A'']A-^A''(y^i](A}foreachA 1,2,3, (here z: ^(C") ^(C"} ^
...
G
501 (g)
is the
^(C") identity
map}. (3)
There exists
a
set
{F} ofoperators
in
=^()
such that R
=
Y^^^
F*Fa
a bounded operator in 501, and such that ^^ V^AV^ G 501 \vhenever A G 9Jl, and a selfadjoint operator H //* G 9JI such that
is
=
y(A)
=
[iH,A]
-
{R,A}/2
+
^ V^AV, a
for
all Ae'm.
To prove that
(3)
=>
(1)
in this
corollary,
one uses
that 7i, 72? ^^^ 73 defined
by
yi(A)
^
[iH,A],
y,(A)
=
-{R,A}/2,
j,(A)
=
^ VIAV, y.
are
all generators of
g<7,
completely positive semigroups, i.e.,
(^)
=
e''"Ae-''", ^"''(^)
=
e'-'^- (A)
E n
and then the Trotter
=
=
e-('/2)^e-('/2)
y"(^^ 7i ^
,
'
Q
product formula, Corollary 3.1.31, implies that y gencompletely positive semigroup. The theory of non-norm-continuous semigroups of positivity-preserving maps has not been developed very much, but there exist some results on quasifree evolutions on the CCR and CAR algebras (see Evans' review paper [Eva 3] for results up to 1980). For other results in this direction see [[Bra 1]], [[Rob 2]] and [Arv 1]. erates
a
226
States in Quantum Statistical Mechanics
The connection between the Duhamel and various
quantities
two-point function of a KMS state theory has been described by Naudts,
of linear response
Verbeure, and Weder [Naii 2]. Ground states, vacuum
energy
äs
described in Definition 5.3.18, are versions of the positiveby innumerable authors since the early 1950s and
states studied
Proposition 5.3.19 is an algebraic reformulation of Standard results in this field. The set of groimd states of a C*-dynamical System was studied by Araki in 1964 [Ära 7], and Sirugue and Testard in 1971 [Sir 1] examined these states äs 00 limits of (T, ^)-KMS states. ^ The notion of passive states and Theorem 5.3.22 is due to Pusz and Woronowicz in 1977 [Pus 1]. ^
The results
convergence of KMS states are more or less folklore, but the of the auto-correlation lower bound appear to be new. (9^ in Example 5.3.27 were introduced and studied by Cuntz in
on
proofs by The algebras 1976 [Cun 2]. An alternative definition means
was suggested by Evans in 1979 [Eva 4]. /7-dimensional Hubert space, where 2 < /i < KQ- Define (9^^ äs the C*-algebra generated by creation and annihilation operators on the füll (unsymmetrized) Fock Space
Let
be
an
oo
m)
e('')
=
r
Evans' definitions
are
on
.
0
then
(9n Then the group i^ where
=
=
(9^J^^(^(9)]),
^00
-C
(9 n is induced from the group
Y(e^^](-]Y(e~^^)
on
(9^,
CX)
r(e'0
=
@(&-e^^]=e^^^ r
=
0
and 7V is the number operator.
(In particular 7V > 0, so the ground state for state.) The algebras On have several interesting properties, e.g., although the weak closure of (9 is a hyperfinite von Neumann algebra in any representation ((9^ is nuclear), 0^ is not the inductive limit of an increasing sequence of type I C*algebras. Example 5.3.27 is due to Olesen and Pedersen [Öle 5]. More refined examples will be described in the Notes and Remarks to Sections 5.3.2 and 5.3.3.
((9oo,i)
identifies with the Eock
vacuum
Sections 5.3.2 and 5.3.3 The
Simplex property of Kß and the coincidence of the central decomposition decomposition of KMS states emerged from studies of the set of KMS states by Araki, Hugenholtz, Lanford, Ruelle, and Takesaki in 1967-1970, [Ära 14], [Ära 15], [Lan 4], [Rue 5], [Tak 3]. The possible breakdown of Euclidean symmetry in the decomposition of KMS states has been discussed by with the
Notes and Remarks
227
Emch, Knops, and Verboven [Emc 1]. Various invariance properties of the type given in Proposition 5.3.33 were initially derived by Sirugue and Winnink [Sir 2]. The absence of breakdown of symmetry in Theorem 5.3.33A was proved by Fannes, Vanheuverzwijn and Verbeure [Fan 4] and will be discussed further in the Notes and Remarks to Section 6.2.6. Connes' for
theory jointness
cocycle theorem, Theorem 5.3.34, is a key result in the classification Neumann algebras and was proved in 1973 [Con 4]. The dis-
von
of KMS states at different temperatures, Theorem 5.3.35, was deby Takesaki in 1970 [Tak 5]. The type III property of the weak
monstrated closure of
5.3.36, little
a
was
work
more
representation of asymptotic abelian Systems, Corollary by Araki and Stormer in 1972 [Ära 16], [St0 5], and with a they could show that the type III part occurring is actually
KMS
shown
type IIIi.
Although Remark 2 after Theorem 5.3.32 has no later applications it is striking because it equates a physically natural algebraic-ergodic prop erty, weak-asymptotic abelianness in mean for all co E Kß, with a purely geometric property, the facial property of Kß. Equivalences of this type occurred in the decomposition theory described in Chapter 4, e.g., " is a simplex if and only if (^, co) is G-abelian for all co e ", but these involved sUghtly more artificial ergodic assumptions, e.g., G-abelianness or G-centrality. Parts of Theorem 5.3.37 and Corollary 5.3.40 can be recovered from Araki's 1964 paper [Ära 7]. The characterization (4) in Corollary 5.3.40 for separable C*-algebras is due to Dang-Ngoc [Dan 2]. The idea of considering more general faces in E^ than the set of ground states, äs well äs characterizations (5)-(7) in Theorem 5.3.38 are essen tially due to Batty [Bat 1] [Bat 2] and he also proves the theorem without any topological restriction on the group G. If S is a C*algebra, F is a closed face in "..3 and co G F he defines P^ äs the projection onto those vectors rj G ^0 such that rather
(^,7r,,(.)/7) These vectors form
a
closed
subspace
property of F, it follows that
P^^n(j^(&]" P^^ abelian. The
P^
G
in
G
U+F
^
7ra;()'',
.
since F is and
a
face.
Using
the facial
is said to be F-abelian if
CD
is abelian. The face Fis said to be abelian if every state in Fis FF-multiplicity of co G F is defined to be the dimension dim
(^S)Batty [Bat 1], [Bat 2] 1.
Fis abelian.
2.
For each
3.
Fis
4. 5.
6. 7.
co
G
proves that the
F,
na^(^]'
following
conditions
are
equivalent
is abelian.
a simplex. Every pure state in F has F-multiplicity l Any factorial state in F is pure. Any two distinct pure states in F are disjoint. .
If coi , C02 are distinct pure states in F then the face generated and C02 in E^ is equal to the convex set {/Icoi + (l /^0<^2; ^^ ^
by
coi
[0, 1]}.
228
States in
Quantum Statistical Mechanics
He proves this without any
separability assumption whatsoever by invoking Integration theory [Ped 3] rather than the more pedestrian measure theory used in our proof of Theorem 5.3.38. In Order to prove Theorem 5.3.38 from the above theorem Batty rehes upon a very useful observation due to Ed Effros and in more rudimentary form to Kastler and Robinson [Käs 1]. Let ^ be a C*-algebra with identity, G a discrete group and T an action of G on 51. Let ^ (8)1 G be the crossed product of ^ by the action T, Definition 2.7.2. Then ^ and G are canonically embedded in ^ (8)T G. If v4 G ^H and g e G WQ iise A, g also to denote the images of ^4, ^ in ^I (8)1 G. Then there is a bijective correspondence between the state space of Pedersen's noncommutative
^^T
G and the set of all bounded functions
State and O is
positive definite
^
in the
GK^^* such that
:
^(e)
is
a
sense
^(g7^g,-)(T~\A^A,))>Q
/,7=1
for any A/ G
^,ö'/
^ G.
The
correspondence
^,(g}(A)
=
is
given by
p(Ag)
p
\-^
p,
where
.
The
correspondence is a homeomorphism for the weak*-topology on E^H^^G and topology of pointwise weak*-convergence on the space of functions from G into 3l*. In particular, if co G " the function ^co(g) <^, for all ^ G G, is positive definite. This establishes an affine homeomorphism between E^^ and the
=
the closed face
F^
of states p
on
^ 0^ G with the property
p(g\ Agi) for all gi, g2 ^
G,^G ^. If
7ip(A}
and
=
FO in
F^
FCO
F^o
=
A G ^H-^
na,(A} is
a
is
face in
FS^, the
exactly
(;7,7rco(v4)/7)
theorem of
p is such
np(g)= U^(g).
a
=
p(A)
state, and
Thus if F is
and hence in
F9t0^G-
projection
onto
is contained in
[R+F.
Batty
are
translated into this
lowing conditions
are
equivalent
One
=
then
p\,^^,
E^^
shows
now
new
Situation,
we
for each closed face F G
2. 3.
F is
4.
Each extremal state in F is
a
co
face in
^
then its
for
co
,^, image
=
G F
that
the vectors r] G E,^^^^ such that If the Statements of the previous
P(,^na)(^]"Pco is abelian for all co G F. {Ti,,,(^)\jU,^(G]}' is abelian for all co
1.
a
see
that the fol-
Fg:
G F.
Simplex.
inf
|co(^'5)-co(^)co(5)| V y V yi
A'eCo(,G(A]]^ 5.
for allyl,5G^. Any state co G F
6.
If coi
and co2
such that
are
weakly clustering
distinct
^
sense
that
=0
{KCJ(^)^U,O(G)}' ergodic
in the
is
a
factor is
ergodic.
states in F then the covariant
representations (ö^^^, TI^^, t/^,) and (,^.,, equivalent (äs covariant representations).
TI,,,,
U,,,^
are
unitarily
Notes and Remarks
7.
If coi and co2 are distinct by CD\ and co2 in
{Acoi
+
(1-^)0)2;
A e
ergodic
E^
states in F then the face
is
equal
to
the
229
generated
convex
set
[0,1]}.
(1) of (1) in 5.3.38 F Theorem condition on immediately special E^j. Since Condition (1) of the theorem is equivalent to imphes PO^ E(ßnfj^(^]"Ea^ being abehan, Proposition 4.3.7, this shows that Theorem 5.3.38 is also true for general groups G. Batty also gives examples of C*-dynamical Systems based on simple C*-algebras where the sets of ground states form 77-dimensional simplices, n=\, 2.... Let (^, T) be a C*-dynamical System and assume that ^ has an identity. For each ß G [-00, oo], let Kß be the set of (T, jS)-KMS states of (5l, T). In particular Ä'+oo(^-oo) is the set of ground (ceiling) states. Define r(^,T) U/?e[-oo,oc] ^jß As P03 < E03 Condition
of this theorem is weaker than Condition
Theorem 5.3.38. The =
=
with the
topology
inherited from the
weak*-topology
on
E^. Further endow
this space with the fibre bündle structure äs a bündle of compact convex sets defined by the map co i^ j8(co) where ß((D) is the ß such that o) G Kß. In order for 0 ^^^ ß\ T^ ^2^ i-^-? the latter map to be well defined we assume that Kß^ n^^^ that the one-parameter group defined by T on ^/3 for any i-invariant closed two-sided ideal 3 c ^ is nontrivial. The bündle r(9l. T) is called the tem-
perature state space of (^, T). It follows from Proposition 5.3.25 that
r(^. T)
is closed in E^, and hence
compact. By Theorem 5.3.30, each Kß is a simplex if ^ G IR and by Theorem 5.3.37 both ^+00 and K^^Q are faces in E<^. The explicit computation of the
temperature state space of (M, T) is in general a very hard task, äs we will see in Chapter 6, but there are a few cases, apart from Example 5.3.37 where it can be
Bratteli, Elliott and Herman [Bra 14] proved that if F is an arbitrary closed subset of [-(X),CXD] then there exists a C*-dynamical System (^, T) such that ^ is a simple C*-algebra with identity and (^, T) admits a (T, j8)-KMS state if and only if ß e F. The C*-algebra ^ was obtained äs the crossed product of an AF-algebra by an automorphism with T the dual action. (A C*-algebra is said to be an approximately finite dimensional C*-algebra or an A.F-algebra if it contains an increasing sequence of finite-dimensional *-subalgebras with dense union. (Particular examples are the UHF-algebras of Example 2.6.12, [Bra 1]). The analysis in [Bra 14] was extended by Bratteli, Elliott and Kishimoto in 1980-83 [Bra 18], [Bra 19]. They proved that each metrizable simplex bündle U ^ ß ^-^ Kß can be realized äs the finite ß part of the temperature state space of a C*-dynamical System (^, T) where ^ is simple, separable with identity and T is periodic with period 271. (The formal definition of a locally compact bündle of compact convex sets can be found in [Bra 19], page 210). This is even proved for the chemical potential-inverse temperature state space in [Bra 19]. The construction can be made such that Foo are isomorphic to the state spaces of two arbitrary simple AF-algebras '^^^ with identity. Furthermore, if Foo are arbitrary faces of the state spaces of ^^^ one can perturb the dynamics T by an inner perturbation P, äs in Proposition 5.4.1, to obtain a new C*-dynamical System (^, T^) such that the finite ß part of the
achieved. In 1979,
230
States in
Quantum Statistical Mechanics
state space is äs before, but the set of (T^, ib oo)-KMS states is affinely isomorphic to FQQ. The paper [Bra 18] concludes by showing that any metrizable simplex is affinely isomorphic to a face in the state space of a simple AF-algebra with identity. Hence Fj-oo can be taken to be arbitrary metrizable simplices (but this certainly does not exhaust all possibilities for F^oc, see, for example, Corollary 5.3.40). For quantum spin Systems, äs described in Chapter 6, there are a few specific Systems where the temperature state space has been completely analyzed. In addition there are many partial results. These matters will be discussed in
temperature
Sections 6.2.5-7 and the associated Notes and Remarks. In this
plices Kß
case
the sim
always nonempty, Proposition 6.2.15, and generically there is a critical inverse temperature ß^ such that Kß consists of only one point for 1^1 < ß^, Proposition 6.2.45, and ß^, -j-oo for one-dimensional Systems, Theorem 6.2.47. This feature is also shared by the only other non-trivial C*dynamical System (^, T) for which r(^, T) is known in detail. This example was analyzed by Bost and Connes in 1995, [Bös 1], [[Con 1]]. Their C*-dynamical System is intimately connected with the Statistical theory of the distribution of prime numbers. It is defined äs follows: Let Q be the rational numbers and are
define
J ^);fl,öeQ,a>0
r-
one
l
b\(\
0
ayVO
b'\^(\
ba'^b'\
a')~\{}
checks that P is
a
(l b\_(\
)\^ a)
ad
group under matrix
~
\(^
multiplication.
-b/a l/a
Let FQ
~
Z be the
subgroup r, It follows from the
=
{(; ;);.z
identity
l
n\(l b\(\ 1\^(l
0
lj\()
a)\0
na^b^Z
~
l
J
\0
a
that the Fo-orbit
through yFo, when FQ acts on the left on F/Fo, conwhen y points cj (Q^)and a p/q with /?, q mutually prime. Since all these orbits are finite, one may define the Hecke algebra (F,Fo) äs the algebra of functions / on Fo\F/Fo, with finite support, with product tains
exactly
=
=
E /()^)^r^)/(ri)
(/*/)(y)yi
where
.
ero\r
/, /' G (F, FQ) are viewed äs FQ- biinvariant functions on F. The algebra ^ is defined äs the norm closure of (F, FQ) in the regulär presentation l on L-(Fo\F), defined by
C*re-
Notes and Remarks
W/)!?)
231
E /(^vr')^(yi)
=
yi6ro\r
The
is
algebra
f*(y) /(y"')on (r,ro) by =
under
closed
The
automorphism
group t,
t/(/)(7)
-
/l(/)*
since
involution,
Sl is defined
on
where
/!.(/*)
through
its action
a'7(7)
if
{l
b
^=^0 fact
In
9}^
FO
=
G
is
T
modular
the
automorphism
for
group
the
l((r, FO))'' given by the unit vector corresponding FO\F, i.e., this state is a (T, -1)-KMS state.
There is
a
natural action
with T, where p ranges
(T,j5)-KMS l <
a
jS
<
00
state
q)ß
over
on
er
all
state
the
to
on
coset
of the group W X p(Z//?Z) on ^ commuting prime numbers. For 0 < ^ < l there is a unique
(^, T) which is (T, j5)-KMS
each extremal
a
type IIIi factor
state is
a
any such extremal state then the map W 3 g\-^ cp
between W and the set of extremal
(T, jß)-KMS
state.
type I factor o
(jg is
a
Moreover, for
state and if q) is
homeomorphism
states. All of these
are com-
puted explicitly in [Bös 1]. Corollary 5.3.41 is related to a famous theorem of Haag proved in 1955 [Haag 4] which gave an important impetus to the C*-algebraic approach to quantum field theory. Let ^ be the C*-algebra generated by the time zero fields in a free, relativistic, irreducible quantum field theory (we refer to the monograph by Streater and Wightman [[Stre 1]] for definitions and a complete acbe the free time evolution, count of the theorem), and let rf ^^
Tf^^(^)
::.
e'^^O^e-^V//o ^
where HQ is the free Hamiltonian and ^ G ^. Now,
another relativistic field zero
theory
fields and time evolution
exists
a
unitary operator V(t)
on
ij^^
assume
that there exists
Hubert space with the Moreover, assume that for each the
same
same
t e
time-
U there
such that
T;"'(^)
=
v(f}^^^%A)v(tr
Haag's theorem then states that the new theory is unitary equivalent with the free theory. The operators V(t) played a large role in the early attempts of constructing interacting quantum fields in the so-called interaction picture, but Haag's theorem implied that the interaction picture only exists when there is no interaction. A program of circumventing these difficulties by using automorphisms of C*-algebras instead of unitary operators on Hubert space was proposed by Guenin in 1966 [Gue 1] and this program was followed in the early days of constructive quantum field theory. Around 1972 these methods were replaced by the more powerful probabilistic methods of Euclidean quantum field theory (see Notes and Remarks to Chapter 6). for all A e^.
232
States in
Quantum Statistical Mechanics
Modular
theory and the theory of KM S states have entered into quantum theory for various reasons some of which are disciissed in Chapter 5 of the book by Haag [[Haa 1]]. First, the relativistic vacuum is separating for the W*algebras of observables corresponding to space-time regions whose causal complement contains a non-empty open set. Hence the field theory has a modular structure. Bisognano and Wichmann [Bis 1]. [Bis 2] established an unexpected identification between the modular automorphisms for wedgeshaped regions and the Lorentz boosts in the relativistic symmetry group. The wedge-shaped regions provide the simplest examples of horizons (see Rindler [Rin 1]) and the Bisognano- Wichmann result has an interesting interpretation in terms of the Hawking temperature of the corresponding black hole (see [[Haa 1]] page 248). In general wedge-shaped regions are the only ones for which the modular automorphisms correspond to point transformations in Minkowski space. Hislop and Longo [His 1] have, however, extended this identification to other regions for theories with a conformal invariance. Secondly, Jaffe, Lesniewski, Osterwalder and Wisniowski [Jaf 1] [Jaf 2] have introduced a super-KMS condition in the context of Connes' theory of entire cyclic cohomology of quantum algebras [[Con 1]]. This concept allows one to deal with situations such äs supersymmetric field theory on noncompact field
manifolds for which the Hamiltonian has continuous spectrum. It relies on the existence of a super-derivation whose square is the generator of a one-parameter semigroup on the quantum algebra. The analytic properties of this su
per-derivation were discussed further by Kishimoto and Nakamura [Kis 6]. This super-KMS condition was partially inspired by work of Kastler [Käs 4]. Thirdly, Bros and Buchholz [Bro 1] [Bro 2] have suggested a relativistic formulation of the KMS condition for thermal equilibrium in which the group of time translations is replaced by the group of space-time symmetries. This Suggestion is based on the observation that the relativistic correlation functions have stronger analyticity properties than those characteristic of the usual KMS
condition.
Theorem 5.3.43
proved by Kishimoto [Kis 4] and he also generalized theorem, Theorem 3.2.46.
was
the Borchers-Arveson
For classical Systems the KMS condition is not
equilibrium because it forces acterization was proposed by
the
dynamics Katz [Kat 1]
o,({f,cj}] Here
ß
=
an
condition of
alternative char-
and has the form
ßcD(g{f,H})
is the inverse temperature, H is the
appropriate
to be trivial. An
.
Hamiltonian, and {, } is the Poisson
bracket:
y f^^
4^ \dqi dpi This condition has been studied tween States
ample,
satisfying
by
dj^d^ _
dpi dqi
several authors and the
equivalence
this condition and canonical Gibbs states been estabhshed in [Aiz 1].
has, for
be ex-
Notes and Remarks
233
Section 5.4.1
UfU-t in Proposition 5.4.1 and Corperturbation expansion for Ff ollary 5.4.2 goes back at least to work in the late forties by Tomonaga [Tomo 1], Schwinger [Schw 1], Feynman [Fey 1], and Dyson [Dys 1], (see also [Ära 19]), while the direct expansion of if was defined by Robinson in 1973 [Rob 8]. The existence of a normal T^-KMS state co^ for the perturbed dynamics T^, and the expansion for the corresponding vector Of in Theorem 5.4.4 is due to Araki
The
=
expansion of co^ in terms of the was derived by Bratteli, Kishimoto, and Robinson in 1978 [Bra 13]. Note that Theorem 5.4.4 is a partial converse of Connes' cocycle theorem, Theorem 5.3.34, and the investigation of CD^ was partially inspired by Connes 1973 paper [Con 4].The notions of per turbed State and perturbed dynamics were extended to some unbounded re lative Hamiltonians by Donald [Don 1] [Don 2] using completely different in 1973
[Ära 20] (see
also
truncated functions of
co
[Ära 2l]),
in the
while the
same
theorem
methods from those of Araki. The
time-dependent approach
to
stability
was
initiated
by Robinson
in
1973, [Rob 8], and Definition 5.4.8, Example 5.4.9, and Proposition 5.4.10 come
from this paper.
morphisms ^.j. are not always automorphisms under the assumptions of Proposition 5.4.10. This problem has been discussed by Golodets [Gol 1] and Maassen [Maal]. Proposition 5.4.13 and most of the remaining results in this section were proved in [Bra 13]. Example 5.4.11 is basically the second quantization of quantum-mechanical scattering theory with a rank-one projection äs perturbation, i.e., interaction. The abstract theory of this problem is described fully in Kato's book on per turbation theory [[Kat 1]] and in particular the M011er matrices are proved to be unitary. This latter result was crucial in the Birman-Kato proofs of unitarity of the iS-matrix for physically more relevant scattering problems.The explicit form of the kernels of the M011er matrices can be found in a variety of places, for example, in Newton' s book on scattering theory [New 1]. The equivalence of normality and finite density for quasi-free states which is used in Example 5.4.15 has already been discussed in the Notes and Remarks to Sections 5.2.1-5.2.3. It is a special case of the criterion for quasi-equivalence of representations generated by two quasi-free states. The
Section 5.4.2
original idea of deriving the KMS condition from conditions of stability purity of a state appeared in a paper by Haag, Kastler, and Trych-Pohlmeyer in 1974 [Haag 7]. This paper assumed L^-clustering of the truncated n6. By exploiting the additivity of the joint 2, 3, point functions for
The and
=
,
spectrum of the Hamiltonian and the modular operator, Bratteli and Kastler were subsequently able to replace 6 by 4 [Bra 12]. However, both these papers
234
had
States in
Quantum Statistical Mechanics
gap in the
proof (Observation 3 of Theorem 5.4.19 was not explicitly This gap was finally filled by Hoekman [Hoe 1]. The proof of The orem 5.4.19 is based on these three papers. In 1977, Haag and Trych-Pohlmeyer [Haag 8] suggested that one could a
proved).
replace the L'-clustering assumption of 4 by simple three-point clustering. ^
the truncated
n-poini functions
up to
=
inf '
r
\ti
lim tj\ ~
co(it^(Äi)i:t,(Ä2}i:t,(Ä3)}
=
co(v4i) (^(^2)^0(^3)
-^00
J
alternatively by
the
assumption that co is a factor state, in the presence of an and they proved Observation l of Theorem 5.4.20 under these assumptions. The proof of the theorem itself was however incomplete. The final Version of Theorem 5.4.20 was proved by Bratteli, Kishimoto, and Robinson [Bra 13]. The stability condition (2b) of Corollary 5.4.21 was also suggested by these latter authors, while the stability condition. or
L'(%)-asymptotic abelianness,
lim
llcoi^
A^o"
=^
coll
=
0
in the
subsequent remarks was proposed by Haag and Trych-Pohlmeyer [Haag 8]. The final remark of this section is due to Kastler [Käs 2].
Section 5.4.3 The
of the chemical potential presented in this section was developed in by Araki and Kishimoto [Ära 17], and Araki, Haag, Kastler, and Takesaki [Ära 18]. We benefited greatly from the lecture notes by Kastler in preparing this section [Käs 3].
theory
1975-1976
Section 5.4.4 This section is based Woronowicz
[Pus 1].
entirely
on
material from the 1977 paper
by
Pusz and
Models of
Quantum
Statistical
Mechanics
6.1. Introduction
previous chapter we examined various applications of the formalism of *-algebras to quantum statistical mechanics. These applications principally concerned equilibrium properties of macroscopic Systems and feil into one of two categories. First, we analyzed thermodynamic limit phenomena of specific particle models by use of the Gibbs ansatz for the equilibrium states. Second, we examined the structure of a set of states, the KMS states, which appeared appropriate for the description of equilibrium. The first of these approaches was developed for the ideal quantum gases in Sections 5.1 and 5.2, while the second was contained in Sections 5.3 and 5.4. It should be reemphasized that the thermodynamic limits of the Gibbs equilibrium states of the ideal Fermi gas are exactly the KMS states of the C*-system formed by the CAR algebra and the corresponding free evolution. For the Böse gas, however, the Situation is more complex because the free evolution T is not a strongly continuous group of *-automorphisms of the CCR algebra M and hence (91, T) does not form a C*-system. Thus, it is not evident that one can use a global C*-structure to characterize the set of equilibrium states. Nevertheless, the Gibbs equilibrium states (D are faithful states of the CCR algebra ^, the free evolution T has extensions f to the corresponding von Neumann algebras na:,(^]" and the (7rco(^)'',T) form ^*-dynamical Systems for which co is f-KMS. Thus, the modular or KMS structure is still a characteristic of the equilibrium states but it is manifested state by state without the synthesizing C*-dynamical System. The aim of this chapter is to extend these investigations by examination of a larger class of models. In the latter half of the chapter we analyze point particles interacting at a distance but before this we describe a simpler, but more artificial, type of model, quantum spin Systems. Spin Systems were first introduced in the discussion of magnetic properties of crystalline substances but have subsequently been reinterpreted äs models of Hquids and gases with restraints on the movements of the constituent particles. There are two advantages in examining spin Systems. First, there is a certain understanding of the collective effects occurring in these models which lead to phase transitions and, second there is a fairly well-developed analysis of the KMS structure, at
In the
,
the C*-level, of these models. The results for continuous Systems are much more modest, however, and even the collective phenomena are barely understood.
Basically there are two directions for analysis, the existence and properties dynamical evolution, and the properties and structure of the equilibrium
of the
238
Models of Quantum Statistical Mechanics
For most quantum spin Systems the evolution dependently of the equilibrium structure but for more States.
cluding
can
be handled in-
general models, in-
continuous Systems, this is not the case. There is virtually nothing known about the evolutions of continuous Systems of interacting particles except in, or dose to, equilibrium and even in the equilibrium Situation knowledge is scant.
6.2.
Quantum Spin Systems
6.2.1. Kinematical and
Basically
a
Dynamical Descriptions
quantum spin System consists of
lattice and
set of
particles confined to a however, two possible physical
a
at a distance. There are,
interacting interpretations of these models, either äs a lattice gas or äs a spin System. The first views each point of the lattice äs a possible site for a finite number A^ of particles, i.e., each point of the configuration space can be either empty or ,7V particles. These particles then interact with one another occupied by l, 2, and this leads to an evolution in which one envisages the particles jumping from lattice site to lattice site. The second interpretation assumes that every lattice site is permanently occupied by a particle but the particles have various internal degrees of freedom, e.g., the particles could have an intrinsic spin with several possible orientations. The interaction between the particles then follows from a coupling of the internal degrees of freedom and this yields, for example, an evolution in which the spin orientations are constantly changing. In some applications of these models it is important that the lattice L has a Z^', i.e., the symmetry. For example one commonly examines the case L lattice is a v-dimensional cubic array, and then the configuration space of the model is invariant under the group of lattice translations. More complex symmetries are also possible. Invariance properties of this type reflect the traditional view of crystals äs objects with a regulär spatial order. There are, however, other crystalline structures, quasicrystals, which appear to lack a precise spatial symmetry. Experimentally quasicrystals are characterized by a discrete diffraction spectrum which is incompatible with any translation symmetry. The determination of the quasicrystal structure from the diffraction data is, however, extremely difficult if not impossible. One possible model of quasicrystals is by a lattice associated with an aperiodic tiling of the configuration space K' but there appear s to be no compelling evidence for any particular aperiodicity. Lattices constructed from aperiodic tilings have a homogeneous structure, but no complete spatial symmetry, and homogeneity rather than symmetry should be the essential feature for any model of a crystal. In the sequel we will see that for some properties such äs the construction of dynamics it suffices to assume that the lattice L is a countable set of points. We assume throughout that L is both countable and infinite. For many results some homogeneity is required .
.
.
=
240
Models of
Quantum Statistical Mechanics
and
we usually assume symmetry and specialize to the simplest although other symmetries or weaker homogeneity properties
/^'
case, L
would often
suffice. The kinematic structure associated with
been described in
Chapters
constmcted
the finite subsets of L.
ExpHcitly,
over
one
a
quantum spin System has already
2 and 3. It consists of
quasi-local UHF algebra (See Examples 2.6.12 ancj 3.2.25.) subsets of L by inclusion. Next, one
first Orders the set of finite
associates with each
point x G I/ an (A^ + with each finite subset A c L the tensor A
=
a
l)-dimensional product
Hubert space
j-^|
and
space
(g)W .xeA
The local C* -algebra ^A is then defined
by ^A 0, ^(A)- If AI n AI isomorphic to the C*-subalgebra AI A. ^AI ÖA, ÖAz ^AI (^ I^AZ of ^(AI AO? where 11A2 denotes the identity operator on A2- Identifying ^A, and ^AI (8) 'B A. one concludes that the algebras {"^IAJACL form an increasing family of matrix algebras whose union is a normed *-algebra, which is incomplete because L is infinite. The norm completion ^l of this union is a quasi-local C* -algebra when equipped with the net of C*-subalgebras ^A (the orthogonality relation J_ between the A is defined by Ai_LA2 if AI n A2 0). The local algebras '^U represent the physical observables associated with the particles at the points of A and the algebra 51 corresponds to the observables of the complete spin System. The quasi-local structure of ^, U^ ^A, is summarized by the following; then
^^^
=
u
=
=
i^
u
=
(1)
^A. C^A2
(2)
^
=
ifAi ^A2
U
^A
,
,
Acl
where the bar denotes uniform
(3)
^ and each ^U have
(4)
identity
D ,
[^A,,Aj=0 whenever AI
(5)
closure,
a common
^ is
n
A2
=
0,
simple.
The first four
properties follow by construction and the fifth follows from isotony of observables and the fourth the quantum-mechanical independence of observables associated with disjoint Subsystems. If L /\ then the group T has an action T äs *-automorphisms of 5l. This action was constructed in Example 4.3.26. Referring to this construction one finds that the translations T have the properties
Corollary
2.6.19. The first property reflects the
=
(6)
(a)
if^
(b)
hm ||[^,T,(5)]||=0 00 |.X|
G^IA, thenT,(^) G^IA+.X,
-^
for
ally4,5
G
51.
241
Quantum Spin Systems The
abelianness follows from conditions
(2), (4), and (6a), and theory ergodic Chapter 4 is applicable to the group Z^ of lattice translations. Physically the group T corresponds to translations in the configuration space and the asymptotic abelianness demonstrates that any pair of observations becomes independent when performed sufficiently far apart. Classical-mechanical spin Systems can be described in a similar algebraic fashion. One associates with each x G L a maximal abehan C*-subalgebra ^{x} C ^(^^x}) ä^d defines the local algebra GA by
asymptotic
ensures
of
that the
states described in
^^-^(.} jce
for each A c I. The
global
A
is introduced
C* -algebra
U
ß:-
GA
by
.
AcL
If L
Z^, the ^i^x] are chosen such that the invariant, i.e., one takes =
^{^+y}
=
group
T
of space translations leaves
'^y^{^}
for all jc, 3; e Z\ EXAMPLE 6.2.1.
Assume N l, then each 9)^^-^^ is two dimensional, ^(;c} is an algebra of 2 x 2 matrices and each such matrix can be decomposed äs a sum of multiples of the Pauli matrices ÖQ(= H), cr^, 03, cr^ defined in Example 4.2.7. Thus if |A| denotes the number of points in A, then ^A is a füll algebra of 2l^l x 21^' -matrices corresponding to polynomials in the Pauli matrices {öf;z 0, l, 2, 3, ;c G A}. A general element of the quasi-local algebra ^ is a function of the cj^ which can be constructed äs the uniform limits of polynomials. If L Z^', one has ixCcrf) cr^^-^, etc. In this latter case the classical algebra is *-isomorphic to the C*-subalgebra ( C ^ generated by the third-component Pauli matrices {0^3; x G //}. This model is often referred to äs a spin-^ System because a quantum-mechanical spin-^ particle has exactly two degrees of freedom. The general case corresponds to N -}- 1 degrees of freedom. spin-A/'/2 because spin-A^/2 particles have 2(N/2) + 1 =
=
=:
=
=
The
foregoing
structure defines the kinematics of the
i.e., the instantaneous observables. We
quantum spin Systems,
next consider the
dynamics
and for this
purpose it is necessary to introduce interactions. An interaction ^ is defined a
äs
function from the finite subsets X (2L into the hermitian elements of 91 such
that
O(Jf)
particles
G ^x- Each <^(X] represents the energy of interaction of the set of all in the finite subset X. Now in the spin System the particles are con-
sidered to be fixed at the lattice sites and hence the total energy of interaction in a subset A consists of the interaction energy of all Subsystems. This total energy is defined to be the
Hamiltonian
H^(A.)
H^(A)
=
associated with A.
Yl
^W
^C A
Clearly, H^(A)
is
a
hermitian element of ^A-
Explicitly
242
Models of
Quantum Statistical Mechanics
An interaction is said to be of finite ränge if the
only
non-zero
con-
tribution to the energy arises from finite clusters of neighboring particles. In Order to make this precise one must assume that the lattice has some rudi-
mentary form of homogeneity metric
a
d(-]-)
following >
r
formly exists
a
for
öfd)
at most
are
jc
symmetry. If, for example, L is equipped with
or
define L to be
we
properties.
two
l there
then
homogeneous
if the metric has the
l for all x,y ^ L, Second, for each First, d(x ; j) finite number A^^ of points y with d(x]y)
a
The interaction > is then defined to have finite ränge if there l such that ^(X) 0 whenever D(X) sup.^^^;^ d(x]y) > Jo.
G L.
>
=
The minimum
value of
possible
=
d^ is called the
mutual interaction between
Ö
ränge of
and there is
no
whose
particles Separation is greater than this ränge. Lattices that are homogeneous include f, equipped with the usual metric, or more complicated periodic sublattices of [R^, and the lattices associated with aperiodic tilings of [R^. If L Z^, or some other subgroup of R^', one can also consider translainvariant interactions. These are the O satisfying the additional retionally quirements =
^(X]
G
ö^;^
C
^A- for all
X
algebra describing
a
C.L, then O is called
a
The set of interactions of
when
with
equipped
classical
quantum spin System forms
a
the
obvious
spin System
and
classical interaction.
linear
a
real vector space
(^\^ 0 is a family of positive functions, one can specify a norm by =
^\(X) ^<^2(X)
and
(/i))(^)
||>||^
=
=
^\\^(X)U(X)
sup
X3X
xeL
and then introduce
a
Banach space
B^ This
method,
always
=
B^
of interactions
{^-m^<^^}
by
.
and variations of it, will often be used in the
be taken
äs a
simple
diameter sical
structure
functions of the number of
sequel. The ^(X} will points \X\ in Z and the -foo often have a phy-
Z\ The conditions \\^\\^ < D(X) of X if L l and one defines interpretation. For example, if ^(X) =
,j(x)
=
^
.
X3X
it follows that
E^(x}
G
^ and
||'CD(;C)||
<
energy of interaction of
at the
Thus this choice of
to a
particles ^ corresponds
||O||^. point
But x
E^(x) corresponds to the with their surroundings. of finite energy per particle.
^ L
restriction
Quantum Spin Systems Note that if
one
considers
each of the Banach spaces This follows because.
and the
sup
^IlOWil^W^^IIOWIItW X^x
is
EXAMPLE 6.2.2.
invariant
^(X)
=
only translationally invariant interactions, then subspace of finite-range interactions. B^ contains the
xeT'
right-hand side
6.2.1. The
243
a
^30
finite sum, and hence bounded.
Let L Z'' and consider the spin-j System described in Example simplest interesting interaction is a one-body and two-body translationally interaction, i.e., ^({x}) T,-O({0}), O({;c, 3^}) Tx-^({0, v -.Y}) and =
=
0 if
l
y^
|>
=
2. One such choice is detemiined
0({0})=/.cr (D({o,x})
by
,
;^7,-W'^^-;
=
,
/:=1
where h
e U and they/ are real functions over Z\ This model is usually referred to äs anisotropic Heisenberg model if the y/ ^ 0 and j/ 7^ jj for some pair /, j ; the 1.2 but Isotropie Heisenberg model if y, j. 7^ 0 ; the X-Y model if ji ^ Q.i j\_ 0. In each case the energy of the system confined 0] the Ising model if y, j\ j,
the
=
=
=
to A consists of two
=
=
=
parts
E .v
E E/x^-j')'^-;'^;'
^^3-
G A
A
.v, 3'
The first contribution
l
/
to the
interaction of the
corresponds electromagnetic field of strength h at interaction, at a distance, of particles x and y. external
x
The dynamical evolution of any System evolution of the associated observables. For A C L, with interaction O and Hamiltonian the
Heisenberg
relations A G ^A
I^(A] i.e.,
r G
[R
i-^T^
algebra MA-
is
We
a
f^
=
T^^(^)
G
spin4 particles
and the second
can a
an
to an
be defined in terms of the
finite
//(D(A), ^A,
with
corresponds
spin System
confined to
this evolution is
given by
where
e''^^*(^)ylg~'^^'J'(^)
one-parameter group of *-automorphisms of the matrix
analyze this evolution for various classes of large number of particles. The problem is by asymptotic expansion in terms of inverse powers of the total number of particles. The first term of such an expansion corresponds to the approximation that the System has an infinite number of particles and this approximation then coincides with the thermodynamic limit discussed in Chapter l and applied to the ideal gases in Chapter 5. Thus, computation of the time evolution of a fixed observable Ä G ^AO consists of calculating limits now
want to
interactions and for Systems which consist of a Standard method of approach to this type of
i,(A}
=
lim A>
00
x^(A)
244
Models of
Quantum Statistical Mechanics
i:^(A}
of the evolutes
äs
the System grows indefinitely in size. We adopt for ^ oc indicates A eventually contains any
the convention that A
simplicity
finite subset of L. There
various
are
possible
senses
in which the limits
T^(^) might
exist and
these govern some of the basic properties of the evolutions i, e.g., the continuity of the map t\-^i:f(Ä), For many classes of interactions the limits exist in the
actually
for all A G ^AQ and all AO C L. Therefore, the by continuity to a strongly continuous one-parameter *-automorphisms of the quasi-local algebra
evolution of
group
norm sense
extends
T
^-
y
2l A
Acl
and
(^, T)
limits
only
forms
C*-dynamical System. For more general interactions the representations and the evolution determines 3. W-
a
exist in certain
system. Our immediate purpose is There
evolutes and
are
T^(^). are
proving the existence of the norm limits of the applies to a slightly different class of interactions different physical feature of the interaction. All known implicitly, or explicitly, dependent upon the equations of
Each method
emphasizes
methods
to describe the C*-results.
several methods of
a
either
motion
JT^
"^^) dt
of
if (^)-
<5A(Tf (^))
Here ö\ denotes the bounded derivation defined
öf,(A) First
=
i[H^(\),A],
^
Ae^i
by
.
examine convergence of the T^ via convergence of the infinitesimal one needs the following version of Proposition 3.2.52.
we
generators ^A- For this
Proposition tisfying
6.2.3.
Let ^ be
an
p,^(x)
=
interaction
of
a
quantum spin system
sa-
Y,mm<+^ X3X
for
all It
X
L.
follows
that there exists
derivation 6
a
of
the quantum
spin algebra ^
such that
D(ö]
U
=
^A
ACL
and for A G
^A
d(A)
=
i
Yl
[i-w-^.
JTnAT^ 0
The derivation ö is
and its closure d is the infinitesimal of a strongly continuous one-parameter group of -automorphisms T //, and only if, one of the following conditions is satisfied:
generator
of
^
norm- closäble
"^
Quantum Spin Systems
245
either d possesses
a
dense
sei
e lerne fits
of analytic
or
(^
aJ)(D(J))-^,
aGK\{0}
the group
if
+
Finally if ö generates
and
i
^A(^)_^/.//.(A)^^-/r//.(A)
for
all yl G 21,
>
^
||T,(^)-Tf(^)|HO
lim A
.
oo
uniformly for
t in
compacts.
First for v4 e ^A
PROOF.
E
i
[^W'^1
and hence ^ is
Proposition
a norm
^ 2
IAIIMII supp
defined derivation. But d satisfies the
densely
3.2.22 and thus ^
are
< +^
;ceA
^9A:
xeA
A'nA7^0
dissipative. Consequently,
hypothesis
of
d is norm-closable
by
Lemma 3.1.14 and
||(z + aJ)(^)||>|M|| for all
G R, and A G D(d], by Lemma 3.1.15. The criteria directly from Theorem 3.2.50. Finally, if d\ is the bounded derivation defined by a
for
J
to be a
generator
follow
df,(A] for all y4 G
\\d^(A]-~d(A]\\=(^
lim
G^(^).
CXD
Thus
lim A*
for all ^ G
Remark.
^, uniformly for
Proposition
6.2.3
t
interaction O. To
see
\\^t(A]-e^^^(A]\\=^
00
in compacts,
has
quantum spin algebra ^ with
implies
i[H^(\),A]
^, then A>
for all y4
=
a
by Theorem
converse,
D(d]
=
any derivation
i.e.,
UACL^A
3.1.28.
can
this first note that the last remark in
that for all finite sets A
c
L, there exists
an
d
of the
be obtained from
J^(A)
=
an
3.2.25
Example ff (A)* 6 ^ such
that
ö(A)^i[H(A),A]
(*)
246
Models of
Quantum Statistical Mechanics
for all A G ^A. Moreover, in
^^^
n
31
^^^
=
.
^(A)
is determined
by ö iip to addition by complement of A and
Here A^ denotes the
2lA.
u
=
elements
^A'
A'c A"
To normalize
Ä(A)
define
we
of
projection E/^
a
norm one
for any (finite or infinite) region A äs follows. Write 3l be the unique normalized trace on the UHF algebra 3(A^ =
from ^ onto ^A (8) 31 AS and let
^A
T
Then E\ is defined
by
^A(^A-^^/) -5ZT(5/Hfor.4/ e ^IA and B i G 31 A^. By replacing ff (A.) by ^(A) (*) and
E^c(ff(A.))
AK^(A)) and conditions
(*)
and
we
may choose
an
^(A) satisfying
O,
=
M
Ä(A) imiquely.
(**) determine
Now, define
//(A) and define
=
^A(^(A)),
r(A)
=
H(A)
-
//(A)
inductively
^(&)
(D(A)
0,
=
=
//(A)
^
-
)(X)
.
;rc A
Then
has
one
//(A)
^
-
Ä(A)
(D(X),
=
//(A)
+ W (K)
.
;rcA
Using
the normalization condition
W(^}
=
one can now
lim A'^
r (A ;
show that
AO
co
where the limit converges in norm, and
r(A ; AO
=
^{^(X\,
To prove this note that if
A,
(^(A))
Here the second
-
//(AI)
equality
AI
Xn A C
l^n A^'
^ 0,
a
1^ c
A'}
.
A, the following computation is valid
^(Ai))
=
A,(//(A)
=
T(//(A)-//(A,))
-
t(A.(//(A)))
-
-
T(AC(//(A,)))
=
0
.
follows from
//(A)-Ä(A,)e2lA,'^2I which is
7^ 0,
consequence of
(*),
while the last
=
älA.
equality
follows from
(**).
(;j
Quantum Spin Systems Next
computes that if A
one
//(A'\A)
=
=
=
where the first two
C
A',
247
then
A,\A(^(A')) A'\A(^(A')-^(A)) E^Ef,^(H(M)
-
H(^))
follow from
equalities
=
(*^)
E^,(H(\') and
^(A))
-
(11)
,
(**), respectively,
and the
last from
^(AV^(A)^^A^ We
derive
can now
for the surface term
manageable expression
a
^(A ; A')
=
H(A'}
-
//(A'\A)
-
.
^(A; A'),
H(A)
E^iff(A'}}-E^>(ff(A')-ff(A))-H(A) ^E^iff(A))-H(A} =
for all A' 5 A, where when A'
>
(**)
lim A'^
The first
used in the second Step. But
was
E^'(A)
-^
A in
norm
oo, for all ^ G ^, and hence
specific
W(A ; AO
=
^(A)
-
//(A)
=
^(A)
.
oo
result
existence of
on
a
first criterion of the above
time evolution is based upon the for
proposition and is applicable to interactions the many-body forces are negligible in a suitable sense.
which
Theorem 6.2.4.
Let ^ be
an
interaction
of a quantum spin system satisfying
the requirement
Mi
=
E^"'f^"pE iiwii)
for
some
>!
>
0, and define
derivation d
a
D(ö)
< +'X)
^
^^^^.&,
"^0
by
^\J^^ AcL
and
^
ö(A]^i
[^(X],A],
^G^A
.
Xr^\^0
Itfollo\vs that D(S) is a norm-dense *-subalgebra ofanalytic elements ofthe of ö. Therefore, ö generates a strongly continuous one-parameter group of -automorphisms i of^ and closure 6
""
||T,(^)- 1,^^)11=0
lim A>oo
for
all A G M,
uniformly for ^A/^\
t in
_
compacts, where
^////,i,(A)^^-/7//cp(A)
Models of
248
Remark.
Less
Quantum Statistical Mechanics
imposed
theless ö remains
Once
PROOF.
then the
P^(x)
growth
making
of
||$(^)||
with
|^|
D(ö) non-analytic,
have established that each element of
we
remaining
may be but never-
follow
Statements
directly
from
D(ö)
is
Proposition
analytic 6.2.3
for ö
because
||0||^forallxGL.
<
A
n
the
on
the elements of
generator. (See Notes and Remarks.)
a
NO W take ^ G T
restrictions
severe
at the cost of
=
^A- One has ^(X)
"^x, and 51 A and ^x
G
commute
whenever
0. Therefore,
ö"(A)
=
E ;^
f
[a)(jr,),[...[(D(xo,^l]]
,
X] n5o/0,...Jr;, n5_i7^0
where SQ
=
\ and
Sj fory
Xj
=
u
Xj-\ u
u
Xi
u
A
+
|A|-7
1. But
>
E-
E E-
=
Xi
and
\Sj <\Xj +|jr._i|
+
...
+
|jri|
.
Thus
Y.
ii<5"(^)ii<2"
n(i+---+"'-i+i^i)ii*iui /=!
i,...,rt>0
where
iii'ii,
=
^^p
E
x^L
x^.^
\X\^i
But fl" <
n
\r"e^^ for A,
a
ii'^wii
+ \
> 0 and hence
P"(^)||<(2/Ari|(D|i:;j|^||;.!exp(A|A|) This establishes that A is
an
.
analytic element for 5 with radius of
analyticity
(V2||||;,), i.e.,
tA >
^^||5"(^)||<+oo ^
>
for
1^1
<
(^/2||(I)||^),
EXAMPLE 6.2.5.
^(X)
=
0 for
\X\
>
and the
0
proof
is
complete.
Assume O contains
sup||a)(W)||<+oo,
-x
=
Z^' and O is
restraint
one-
and
two-body interactions, i.e., are
then valid whenever
sup^||0({x,>^})||<+oo
x&L
If L
only
2. The conclusions of Theorem 6.2.4
.
e I v e L
translationally invariant,
these conditions reduce to the
single
Quantum Spin Systems
E
ii
+
^
249
.
Q^y ^V
This is
obviously
understood
äs a
satisfied if O is of finite ränge and in general it on the ränge of the two-body interactions.
can
be
restriction
The evolution i constructed in Theorem 6.2.4 can be considered äs a function of the interaction $ and the perturbation estimates used in the proof estabhsh that T is continuous in O. For example if Oi and 2 both satisfy the
requirements of the theorem, and if ^CD, corresponding groups T^^ and T^^ then
Tf'(^)-T^(^)=^^
and
Y,bl^(b^,-b^,}bl^"'-\A}
^'
n>\
m
.
Q
=
Hence for A G ^A the above estimation
l|Tf'(^)-Trn^)ll <E
b^^ denote the generators of the
procedure gives
'"(T)" Eil'i''liril*i'2l"-"'-' V'^/
>1
x||$,
m
=
0
-
<^||
whenever
follows
(2|^|//i)||O/||;
<
by approximation
all A and
t
l,/ of ^4
1,2. Continuity for general ^4 and small t then by strictly local elements. Finally, continuity for
results from the group property of t^-^if, concerning the existence of dynamics
The second result
erties of the energy interaction This energy is defined by
the
across
emphasizes propsurface of each finite subset A C Z.
^
ffo(A)=
(^)
A-nA^0 A-
n
A'' 5^ 0
where A^
again denotes the complement of A. The following theorem, which is 3.1.4, covers two cases which are typically applicable to one- and two-dimensional Systems respectively. a
consequence of the result of Section
Theorem 6.2.6.
Let ^ be
an
interaction
of a quantum spin System satisfy ing
the requirement
p^(^)
=
Y,mx}\\<+^ X^x
for
all
jc
G
L, and define
a
derivation d
D(6)
=
by
U Acl
^A
Models of
250
Quantum Statistical
Mechanics
and
d(A)
^
i
=
mX],A],
A^^^
,
X^^^i^&
Assume that there exists
increasing
an
A
seqiience
C Z/
such that A^
>
oo
and
E
either
^(^^
<M
Xr^^n^^0
A'nA^j^0
for
M > 0
some
or
/ X
n
^0
.YnA^,
for
^!2
0.
some a >
It
0()e-""
<
^
A
that the closure d
follows
parameter group
of d generates a strongly of -aiitomorphism ofü and,furthermore,
continuous
one-
*
||T,(^)-Tf"(^)||=0
lim n^r oo
all A G ^,
for
theorem, when
Sn,m
=
t in
compacts.
This result follows from Theorem 3.1.34, and the discussion
PROOF.
the
iiniformly for
^+/,o
=
^n+m,
one
makes the identifications X
=
^^ Xn
=
subsequent to d^ and ^A, S =
where
dn(A)=i[H^(\n),A] for all yi
G
^.
Thus under the first
assumption
of the theorem the result follows from the dis
cussion at the end of Section 3.1.4 because
II^U, -^,o||
=
||(<5-.5)k
E
||<2
^(X]
<2M
.
A'nA^0
.rnA^-^0
Similarly,
the second
L,,-5,,||
assumption =
of theorem
implies
that
E
||(c)-5+,)U^J|<2 .r .V
n
n
(^)
=
0(n)e-^'"
/ 0
A
A'^ ,
^0
and the result follows from Theorem 3.1.34.
Theorems 6.2.4 and 6.2.6 both sufficient for the construction of
imply
give conditions on the interaction which are dynamics. The conditions of Theorem 6.2.4
that
sup||(D({x})|| jceL
<-f-oo
,
251
Quantum Spin Systems
i.e., the interaction energy is uniformly bounded on the lattice. Theorem 6.2.6 does not impose this type of boundedness and ||O(^)|| could grow indefinitely provided that the surface energies are appropriately bounded. Thus, Theorem 6.2.6 appears more general than Theorem 6.2.4. Unfortunately, the bounds of Theorem 6.2.6
are
rather restrictive and
two-dimensional models. These
essentially Hmit the result to one- and applications are illustrated by the following
examples.
EXAMPLE 6.2.7.
Let L
=
Z, A
and
[,],
=
assume
that O is
a
translation in
variant interaction. It follows that
E
<2^D(X)\\^(X)\\
i'W
.
^90
Xr^\^0
A'nA^/0
Thus if the
right-hand side is finite the first case requirement differs from that of Theorem 6.2.4
||0||
=
E
of Theorem 6.2.6 is insofar the
applicable. possible growth of
This
li^'WII
X3Q
\X\=n+l
with
n
is less restricted. On the other hand, the ränge of O is more restricted. For if O is a two-body interaction, i.e., if ^(X) 0 for \X\ > 2, then the con-
example,
=
dition becomes
^|x||0({0,z})||<+oo
Z^ x^Q
.
Thus, the two-body interactions should decrease roughly
EXAMPLE 6.2.8.
Let L
A, and
assume
E
that O is
"^W
A'nA5^0 n A^ 5^0
A-
But the
-
-
{x-x
an
|jc|
-2-g
Z^ =
(xi,X2)
e
Z\ \Xi\
1,2}
,
interaction with finite ränge d(^. It follows that
E
^
äs
E*^w
<
4Ö?(D
SUp
xeA\A;,-rf^
xA\A-rf^
Ew X^x
D(X)>m
right-hand side vanishes identically applicable.
if
m
>
J(D and hence the second
case
of
Theorem 6.2.6 is
The third and final method of
Systems applies only
to
a
discussing
the
dynamics
subclass of the interactions covered
but it establishes
of quantum spin Theorem 6.2.4
by
more detailed propagation properties. This method only Z^, and differs from the previous methods regulär lattices, e.g., L insofar it establishes the Cauchy convergence property of AI-^T^(^) directly. The starting point of the method is the estimate
applies
to
=
252
Models of
Quantum Statistical
Mechanics
d
|T^(^)-T^(^)|
^^^(^^^f-y^)) rf.T^([//o(A,)-ÄI,(A2),Tfj,(^)]) 0
kl Y~^
E / ds\\{<^(X),^^^-(A]]\\
^
^
.YGAl\A2 X3X^^
which is valid for A2 C AI, and A one
estimates this
^A,. Thiis
to prove the Cauchy property occurring in the last summand. These bounds on propagation velocities. We discuss G
estimate the commutators
must
be understood
can
äs
Interpretation subsequently. 6.2.9.
Let >
T and
assiime
Propositioe lattice L
Ill-ll,
-
be
interaction
an
of a qiiantum spin System
on
the
that
^ |X|(yV+ lfWe^W||(D(X)||
sup
< +00
.T6Z^'.r9x
for X
/l > 0. Let
some
eZ^
\-^
i^ denote the evolution assodated with and A
the action
TY
ItfoUo\vs
of space
Kx(c),^] \\c\\
sup
.,t?vC^il{0}
all A
e
Z^' and
that
[Tf(^),5]||< \A\\Y, for
C
translations.
^10}
^^d B
e
^l
,-|.T|A + 2M||cD||,
.
First define
PROOF.
C,%(,r,0
=
[T,T;^(^), 5]
and remark that
C.%G^,O-C^,^G^,O) +
|'J.^C^^,CT,.)
^c^.^(x,o)+/^
rj.[T,Tf([(D(x),^]),5]
.
r^n^O Next define
Cß(x, t)
\\C^,B(^,t)\\ sup
=
sup
Acr' Ae
The
integral relation for C^^, yields the integral inequality
CB(x,t)
<
CB(X,Q)
v-
+
\
and several
f'''
/
XBO-fO
Next
we
seek
a
majorization
UI{o}
of the
applications
triangle inequality,
\\[^^T^([^(X),A]),B]\\
äs sup Acz^'
of the
sup
^e^i(o}
integrand
IMII
in terms of Q.
Quantum Spin Systems Let
has
e(ix, jx], ixjx
0, l,
=
.
.
unique decomposition
a
.
A'', be
,
a
set of matrix units for
[O(Jr),^]
of the form
Y. ca>({/.},{Ä})n^(w.)
[^(x],A]=
^e^
{/.}{/.}
with coefficients CCD e C
Each
^{x}-
253
satisfying
|c*(0.},a})|
\mx],Ä\\\
<
2||
<
.
Substitution of this
decomposition into the integrand of the above inequality, use of linearity properties of commutators and the triangle inequality then gives the new integral inequality the
CB(oc^t]
+
Y. Y.'^(N-^\?^^^mX]\\ ^0r
X^Q
But this takes the
simple
dsCB(x + y,s]
.
y&X
form r\t\
CB(x,t)
<
C5(x,0)
+ 2
V (y) yer
/ J^
dsCB(x-{-y,s)
,
where
w
=
E(^+i)"'"ii*wii A-BO
X^y
Iterating
this last
inequality yields
CB(x,t]
<
^v^^/ö(.-,)^2Me(ö) / ^ ^(3^,0)-!-, (27r) 7o<0,-<27r
^
^T^v where
e is the Z^-Fourier transform of s. (This use of Fourier transformation is where Z^ is important.) As e is analytic and periodic one may regularity assumption L change the integration to an integration along Im ö 7 to estimate
the
=
=
J_
/'^vg^/0(x-^)^2WS(fl)
l
(2. (27c) J
/'rfV^|^2WI(e+,|
(In) J
But
K(> +
iy}=^e-'^'>^'^^ys(y)
.
yel"
Hence, using s(y)
>
0,
one
concludes that
\8(9 +
iy)\
=
B(iy)
.
yez^'
Consequently
|g2|/|(0+/| Combination of these estimates
CB(x,t)
<
immediately yields
the bound
Y, C5(y,0)e-(^->')^e2|^l^('^) yer
But
choosing
y
=
((x
y)/\x
y\)^
with A
>
0
gives
.
_
254
Models of
Statistical Mechanics
Quantum
e(^
Y^ ^(A^+ l)-'^'||)(X)||e^-(-^->')'^'/l-"--^l
=
XBÖ
yex
V|^|(^+l)-W||
<
/
.
^
X^O
Consequently
^Q(>',0)e-'-l->'le2|'IWb-
Q(x,0<
.
y&r'
The Statement of the
Corollary
6.2.10.
proposition
now
//
C(.,.)=sup C(;c, 0)
20^^
<
"[^^^'(^)'^]ll \\A\\m
sup
AcT
then
follows from the definition of CB-
A,;7^,,
and hence
C(^,0<2exp{-|f|(AW/|^ -2||(1)||,)} These estimates
'
allow
now
a new
discussion of
.
dynamical
evolution for
a
suitable class of interactions.
Theorem 6.2.11.
lattice L
=
||(D||^ some
A
>
interaction
an
sup
^
Z'
G
0. Further let
on
the
strongly
to
a
one-parameter group
^
of^-auto-
||T,(^)-Tf(^)||=0
cxD
^, and the convergence is uniform for
Moreover, for A^B
< +00
T^ denote the evolution associated mth ^ and A.
lim A>
all A
quantiim spin System
^ \X\(N^l)^\^^e^^^(^^\\^(X)\\
foHows morphisms T of^, i.e.,
for
a
JTB.r
that T^ converges
It
of
assiime
.T
for
be
Let )
Z^ and
^{Q}
one
\\[i:^it(A),B\\\ l
t in
compacts.
has
^ii^ii ||^||g-klUW/kl-2||a)||,}
||[T,T,(^),5]||J" where
x
G Z^
H^
Finally if^\ the appropriate
TX
denotes space translations.
and ^2
are two
aiitomorphism
interactions
of the
above class andi^^^ andi^-
groups, then
||T*'(^)-Tfn^)ll1-02|| '
e2k| ||0||,
_
l e
II*L
A-/ jez^
255
Quantum Spin Systems all A e
for
^vhere
^A,
||O||,
min(||Oi|| ||O2||;j
-
bc-10 n
x-y\
mm
=
and
y^\Q
Prior to
PROOF.
6.2.9
Proposition
established the estimate
we
r\A
E/ X3x'^^
Y.
||Tf'(^)-T^(^)||<
A:eAi\A2
whenever A G matrix
units,
with AQ C A2 C AI
^AQ
in the
äs
proof
ds\\[^(X].^^'~(A]]\\
.
^||a)(jr)||(7v+ifi^i
||tf'(^)-T^(^)||< ^
X3X
jceAi\A2
'^'
V^ / ^-^Jo ^JQ
X
Therefore,
an
application
\\[^y(B],^^^-(A]]\\
ds
sup
ll^ll
Äe^r^o}
of the bound of
Proposition
xY" y^e-^^i^-i Z^ Z^ yeX -
z|
<
-
z
-
jc|
+
A:
-
and
>;|
||Tf'(^)-Tfn^)ll<2|M||
gives
^^x
xAi\A2
-|3^
6.2.9
^(7V+lfl^l||
||tf'(^)-T^(^)||< ^
But
in terms of
But then decomposition of ^(X] Proposition 6.2.9, yields the estimate
of
|;c
zGAo -
^ a:eAi\A2
j;|
<
/%.^^ii^ii^-
.
Jo
D(X)
for jc, y e J^. Thus
one
obtains
^|Z|(7V+lf'^l||^(Jr)||e^-^(^) ^3^
y^-^^i-^i ^0/'" ^56^11^"^zt^o <|M|||Ao|(ß''^lll^ll^-l) 5^ e-^-Wo X
.
x6A,\A2
Hence
uniform
T^(.4)
on
converges in
norm
for all
strictly
local A and the convergence is
compacts. Convergence for general A follows by continuity.
The second Statement of the theorem follows
directly
from
Corollary
6.2.10.
The final Statement follows from the estimate
||T*'(^)-t^(^)||
/'"rf5 ^ ||[0,(^)-a.2W,T^(^)]||
<
70
and
an
argument similar
EXAMPLE 6.2.12.
body
xcZ"
to the above.
Assume 0 is
translationally
invariant and contains
only
^||a)({o,^})||^^^w
for
two-
interactions. The condition of Theorem 6.2.11 then reduces to the condition
some
A >
0, i.e., the interaction
must decrease
exponentially
with distance.
256
Models of
Let
Quantum Statistical Mechanics
consider in
little
detail the
significance of the foregoing corresponds to an observation at the origin and TYT/(^) can be thought of äs an observation at the point x at time t. The commutator [i:,cit(A),B] with B E ^{o} then gives a measure of dependence of this observation on the observation B at the origin at time t 0, i.e., the commutator measures the effect of the observation Tx^t(A} at the origin. Thus, the estimate US now
a
commutator estimates. If ^ G
more
^{o},
then A
=
['-'<^>'*l<exp(H,|ffl-2||*l|, 2IMII llll shows that this effect is
exponentially decreasing
with time outside of the
cone
/2||$||A
W
.
A
Thus
physical disturbances propagate K.
with velocities less than
,f^
=
A
A
Another way of arriving at this same qualitative conclusion is to reexamine the essential localization of T^(^). The estimate used in the proof of Theorem 6.2.11 shows that
IK(^)
rf (^)ll
-
<
Mll(e*!l^-
-
g^^-W
X^
1)
.reA"
<2\\A\\m,\t\Y,e,reA"
for A G
^{0}.
But if A
=
{x; |;c|
V e-^\^\
<
<
S,
+
D},
l
dr
^v-l^-A.
Jr>V\t\+D~l
xeA^-
where S^ is the surface
V\t\
then
of the v-dimensional unit
area
sphere. Therefore,
||T,(^)-Tf(^)||<2|M|| m,F(\t\]e-^-\'\(^-^>^ where V),
=
2||
and
F(\t\)
F(\t\] where PV-I is
a
polynomial
=
has the form
\t\P,..,(V\t\+D)e-'^\'\
of order
v
1.
Thus, if
,
K >
F(D and
e
>
0,
one
may
choose D such that
|K(^)-Tf(^)|
t G
[R, i.e.,
it(A)
is
essentially localized
|;c| This conclusion that the
damped
outside of
a
causal
<
V^\t
propagation cone
4-Z) of
determined
in
a cone
.
physical effects is exponentially by a group velocity V^ is essen-
restricted to interactions which
tially
Quantum Spin Systems
257
in
config-
exponentially decreasing
are
uration space (see Example 6.2.12). If the interaction decreases more slowly, then one would expect the damping to be less important but no quantitative results in this direction have been obtained. these results show that
Although
propagation
takes
place only
in
a
causal
with asymptote determined by FO the method provides no information concerning the distribution of effects within the cone. The characteristics of this
cone
distribution
much
depend
intimately on the details of the interaction. For velocity of propagation is sensibly smaller typically zero for classical interactions such äs the
more
certain interactions the maximum than the estimate FO and it is
Ising
interaction.
EXAMPLE 6.2.13. A G
^{;c}
Consider the
spin-| Ising model introduced in Example 6.2.2.
If
is of the form A
=flof7j+ 0(3(7^
with üQ^a^ e C, then the commutation relations [0^,0-3] 0 and the identification 1 that show is A fixed for the a evolution, i.e., point Ising ÖQ =
r,(A)=A Altematively
if
A' where a
e
C and
0^
^^^
the
=
one
a^o^^
+
spin raising
cr'^ then
.
=
a-0^_
and
K/a^2)/2
,
lowering operators
defined
by
.
finds
T,(^0
fl+6'^^^4
-
+
fl_e-^''^V_
,
where
^*=E^'3(^->'X yeZ'^
Thus the time
development t\-^'T:t(A')
0 has finite ränge. In this latter ränge of
O)
of the
case
has
T/(^)
a
multiperiodic
nature which is
remains localized within
point jc. Einally a general element of ^{^^ cases gives the generic behavior.
a
periodic if (the
distance d^
has the form A-\-A' and
combination of the above
The
are
fixed
model illustrates
a behavior typical of classical interactions. If spin algebra and O takes values in , then all elements of ( points of the corresponding evolution. This follows because Cf i s
Ising
( C ^ is
a
classical
abelian. More interesting types of propagation are expected for models such äs the Heisenberg model and the X-Y model. To analyze the dynamics of the latter we
first describe
spin-^
a
connection between the kinematics of the one-dimensional
System of Example 6.2.1 and the corresponding CAR algebra of
Theorem 5.2.5.
258
Models of
Quantum Statistical Mechanics
EXAMPLE 6.2. 14A
spin-^ model with L 1. The C*-algebra of local generated by polynomials in the Pauli operators {ö-J; z 0, l, 2, 3, :c G A} and the corresponding quasi-local algebra 51^ is isomorphic to the UHF-algebra (8)?l_ooM2. An explicit matrix representation of the Pauli op erators was given in Example 4.2.7. They satisfy the Pauli relations observables
Consider the
=
is
^U
(cr;)^
=
i,
0-7
0-;,
=
O;CT;^,
=
-O-]^,O;
.V;^,
=
where addition of the indices is modulo 3. Moreover, [öf , NO W define the Fermi algebra ^^ äs the C* -algebra
crj]
{fl^,fl*;jc
G
Z} satisfying
0 if ;c 7^ y. generated by elements =
the anticommutation relations
X,a*}
=
d.,^y
{a^^.ay}
,
0
=
was established that ^^ is also isomorphic to the UHE-algebra (8)Jl_oo^2 but due to the anticommutation at large distances there cannot be an isomorphism between ^^ and ^^ which respects the^ spin structure. Nevertheless one can embed ^^ and 51^ in a larger UHF-algebra Ul in such a way that ^^n 31^ is the C*-subalgebra of even elements, i.e., the C*-subalgebra gener
for all jc, j; G Z. In Theorem 5.2.5 it
ated
by
even
in the
polynomials
a
and
a*,
or
in the
The
a.
embedding
is defined
äs
follows. Let 0_ be the
automorphism, of r\
f
-(-^) The crossed
order 2, of
f
\ =
if
<^x
{-a,
^^ >
X
defined
by
l ,
if.io
.
31 Q_ Z2, of ^ and this action of ^2 product ^ Z/2Z, is then the generated by ^^ and an operator T G ^ satisfying TT* and H, r TA &-(A)T for ^ G 51^. One can check that S is also isomorphic to (8)?^_^M2. Now realize the Pauli algebra ^^ äs the C*-subalgebra of ^ generated by =
=
C* -algebra
==
=
=
ö-j
-
TS^(a,
+
a*)
,
(r\
=
iTS,(a,
-
a*),
er]
=
2fl>.,
-
^
where
fnr=:^3
if-x>l ,
S,=
ifx=\,
[n;=.x<^3 The operators then
^^
into
one
satisfy
the Pauli
ä. (The significance
spin
if^
.
relations and this defines
of the operators TS^
an
embedding
of
S^T is explained by noting that
has the formal relation
Ts.=
n V=
Next define
an
0(z,7)(x)
o^.)
co
action 0 of the group Z2 -
(-l)Vx, Ö(/,7)K)
=
x
Z2
on
^
by
(-l)X, ^(iJ](T]
=
(-lyr
for all /,7 denote the four spectral subspaces of 3t defined by this 0, l, and let ^ action (see Section 3.2.3). Then one checks that 31++ is the even subalgebra of 31-F =
alluded to in the
previous paragraph and
259
Quantum Spin Systems
5I^-ä++
^-+
+
,
^^:^ä+++ä__ The
^^r^ 51^ subalgebra 5I++ interesting automorphism a of
even
An
ö-K) One checks that to
^^
a
jc+i
=
^^
restricted to
note that
^^
is also
satisfy
ö^, ö^
commute if
^
x
is the twosided
now use
for
the
.
shift and its restriction
quasi-free on
the
subalgebra. lowering operators
even
and
spin raising
=
KX-]
1,
üx.a*
TSxO^_,
=
=
TS^o^.
description of the kinematical development of the XY model.
that
ji(x]
(^ for
=
structure of the
1.
|A:| 7^^
cr^ by
.
Consider the Jt^-7-model of
simplicity,
.
be recovered from the
can
al
=
this
the time
EXAMPLE 6.2.14B assume,
(8)^_^M2.
Ki<4)/2
=
y. The
a^
analyze
to
the relations
KX-}
We
(T\T =Ta\
=
generated by
<4
The
^(^)
,
isomorphic by
^ is defined
is the twosided shift. Thus these shifts coincide
Finally
which
is itself
=
.
6.2.2 fo
Example On the
spin-|
algebra
System
v
-
l
to
but
^ the local
Hamiltonian then has the form
//=//a,([-,l)
=
-j|^((l+yKali+'
+
(l-7)<^,(T^+')
^ al
+ Ä
xn
x=n
n-\ =
"^
2J
ö;c+i
+
ö*+iß;c
+
7ü!>J+i
+
yax+\a^)
x= n
y;(2ax-ii)
+h
.
Z-^
x= n
Therefore the Hamiltonian H^ defines
subalgebras ^^
and
finitesimal time
change
of the
lim
/[//, J
<5a)K-)
^
^^
a
time
development of ^ simplicity, given by
invariant. Assume, for a
is then =
-2zJ(ür;c+i
+x-i
which leaves the two that /z
=
0. The in
+y(fl*+i +ö*_i))
/J> OO
For h
=
(f,g)
with
f,g^ hW
B(h}
=
a* (/)
it is
convenient to introduce
now
+a(g)
=
Y. M + Z ^J'^J J
One then computes that the derivation
^,(B(h-)]
d^ generates the one-parameter group
=
B(e'^>^'h]
where
/
(U
+
W]
\-j(U-U^) and
'
j
y(U-W] -([/ + [/')
260
Models of
Quantum Statistical Mechanics
(C//)^. The time evolution of T
F,
^ is
given by i:t(T)
=
=
/,-!
.
TVt where
Te^^^^Te-'^""
lim
-
G
([/*/),
/;+,,
=
n^^oo
r dt, r
^g,-^
^/^ö-(^.)^-w.
lim
=
f^Q
-^^
r^ dtn^,^(A]...i,,(A)
dt2...
Jo
JQ
Jo
,
by Theorem 3.1.33, with A
=
lim
(0-(//)
//)
-
=
-4J(aQai
+
a\aQ
+
y(aQa\
+
ßi^fo))
AZ> OO
These formulae
tion,
one
form h G
the
give
4.
multiplicity
/2(Z)
time-development on Ut. Next, using Fourier transformaa Lebesgue spectrum on [2, -2y] u [2y, 2] with uni e'^^^/z Therefore 0 for all 00 weakly äs t
deduces that K has
0
/2(Z).
^
>
Hence the limits
&(A)
lim
=
MF;
r^00
31^
exist for all ^ G
and define
automorphisms
e.,(B(h))
=
of
51^.
In fact
B(U^h)
where ^
are unitary operators which can be computed explicitly. Using these observations, and a method similar to that of Example 5.4.9, one nojw establishes the following twisted asymptotic abelianness for ^,5 G 21^ : if ^,^ G ^++ then
lim
||[^,T,(5)]||
=
0
,
;^oo
if .4 G 2I__ and B G
^++ then lim
\\Ai,(B)-e+(i,(B))A\\=0
,
^^00
and if ^ G
ä++
and B G
ä__
lim
then
\\Ait(B)-i,(B)^+(A}\\=0
.
/> 00
These
general conclusions^remain vahd if /z 7^ O^Note that it is also possible to show automorphisms 0 do not extend to ^ (see Notes and Remarks). A special case which is illuminating is the case 0, i.e., the isotropic X-Y 7 model. Then the infinitesimal time change is given by that the
=
(5$) and this
can
be
-
\im^i[Hn,a^]
integrated by
=
2iJ(a^_,
+
a^^j)
Fourier transformation to
T,K)
=
+
2^^
give
5]G,(x-j)a; .veZ
where
Gt(x)
=
(2nr^ r\0e^--o^^t(4JcosO+2h} JQ
Thus if ^ But
one
=:
a^QÜQ
=
finds that
a^d^
=
\Gt(yt)\
(d^
+
1])/2
decreases
then
||[TxT^(^),yi]||
exponentially
with
t
is if
proportional
\V
>
4|J|
to
\Gt(x)\.
and decreases
261
Quantum Spin Systems like
|f|~^/^
if
1^1
<
4|J|.
Within the
cone
\x\/\t\
<
4|J|
there is
essentially homogeneous and decreasing proportionally cone the propagation effects are exponentially small.
to
propagation
\t\~^^'^
which is
and outside the
Throughout this subsection we have analyzed the time development of quantum spin Systems äs a norm limit of the local dynamics. For many interactions, e.g., long-range interactions, it is possible that the appropriate limits only exist in certain representations. (See Notes and Remarks.) The periodic time behavior exhibited by the Ising interaction in Example 6.2.13 is easily seen to be typical of classical interactions and of course all elements of the classical subalgebra ( are left invariant. On the other hand it is not at all evident that the propagation exhibited by the J^-7model in Example 6.2.14 is typical of quantum interactions although this is believed to be the case.
6.2.2. The Gibbs Condition for
Equilibrium
Throughout the rest of this section we analyze the equilibrium states of quantum spin Systems and we often assume that the dynamics is given by a strongly continuous one-parameter group T of *-automorphisms of the quan tum spin algebra ^. In this paragraph we compare the traditional Gibbs definition of equilibrium and the definition through the i-KMS condition. Subsequently we discuss other characterizations of equilibrium. First define the local Gibbs state associated with the interaction 0 by
,._Tr5>-^(^
'^^^^^ for all ^ G ^A. and then
corresponding we
Occasionally
we use
often omit
Tr5,(e-^*(A))
it is useful to indicate the
co^.
the notation
to the inverse
"
dependence
It is also natural to insert
a
of COA
on
O
factor
temperature, in front of //<E)(A) but for
jS G IR, simplicity
ß.
Although COA is defined äs a state on ^A, i t has an extension to a state on ^ by Proposition 2.3.24. Let co^ denote an arbitrary such extension. The theroo can now be analyzed with the aid of the co^. modynamic limit of COA äs A The spin algebra 91 contains an identity and hence the state space Esa is compact in the weak* topology. Therefore, there exist nets of extensions co^^ of oo and co^^ converges in the weak* topology to a state co coA^ such that A ^>
^
over
9l, i.e.,
limco^J^) for all y4 G 9l. Hence, in the
sense
co
is
a
-:co(^)
thermodynamic limit point
that
(D(A)
=
limcoA,,(^)
of the local Gibbs states
262
Models of
Quantum Statistical Mechanics
for all A G ^A and all A. Henceforth
thermodynamic a
net of
It is
extensions of
a
dynamic
we
define
a
limit point of the local Gibbs states
co^
6.2.15.
over
^ to be
a
if it is the weak* limit of
to ^.
consequence of the general analysis of limit points co are KMS states.
Proposition
CD^
state
co^
be
Let >
an
interaction
Chapter
5 that the thermo
of a quantiim spin System,
o}'^
the local Gibbs states
Tr,,(.-/^^(AU) ^/^.,,^ ^ ^ '' Trg,(e-/'i'(A))
and f^ the local
automorphism
r^(A)
Assume that r^ converges
strongly
for
all A^^ and
t e
Itfollows State over
to an
v4 e 31
.
automorphism
group T, i.e.,
||Tf{^)-T,(^)||=0
lim A>
'
groiips
e''..(A)^e-'W*(A)_
=
^^ ^A
'
oo
[R.
that every ^.
thermodynamic
limit point co^^
ofco^
is
a
(T, ß)-KMS
As ^ is separable w^"^ is the weak* limit of a sequence of extensions of the local Gibbs states. Therefore the result follows from Proposition 5.3.25. PROOF.
This states
proposition shows
are
KMS states and
that the our
thermodynamic
aim is to derive
limits of the usual Gibbs
a converse
of this Statement and
thus deduce that the Gibbs
equilibrium states are completely characterized by the KMS condition. It is, however, overly optimistic to expect a strict converse to the proposition. The difficulty is that the foregoing notion of Gibbs state is too restricted. The state coy^ describes a closed System, i.e., a System completely isolated in A and lacking interaction with the exterior. But external interactions, or interactions across the boundary of A, are of relevance whenever multiple phases, i.e., multiple thermodynamic equihbrium states are possible because these interactions inüuence the relative proportion and disposition of the phases. Thus, Variation of the external interactions is necessary if one wishes to describe all equihbrium distributions. Hence, it is necessary to broaden the notion of Gibbs states to allow these external effects. There are two approaches. Either one may modify the interaction near the boundary of A in Order to
reproduce the boundary effects or one may consider the System äs a Subsystem of a larger System. Although these methods are conceptually different they are technically similar. We adopt the second point of view and refer to the Subsystem äs an open system, i.e., open to external interaction. We next introduce
a
set of states
which describes equilibrium for open Systems. first examine the Gibbs state co^/ of a finite spin System If A c A^ is a Subsystem then the restriction of the Gibbs
For orientation let
with interaction
ß
us
263
Quantum Spin Systems
MA is
State C0y^,' to
particles
in A
not the Gibbs state CL)A of
interacting
with
^A. The restricted
in the external
particles
the state C^A' is modified by removal of the interaction and A'\A, then the restriction of the modification is express this
quantitatively
sense
across
A7^0,A^n AV0,;^C A'}
the border of A and
Now subtraction of
of the state co^ in the
a
=
Tr^^, (^-/i//.(A)^-^//.(A'\A)^)
Tr^, (^-^^a,(A')+/^^)
^
A^\A.
,
perturbation corresponds perturbation P ßW^(A ; A'). Explicitly to
of Section 5.4.1 with the
A'(^) A'V )
region A^\A. If, however, across the boundary of A equal to COA- In order to
reintroduce
^ci)(A;AO=^X]{^W;^^ the energy of interaction this external interaction
state describes
_
_
= ""
"~
Tr ^(e-^^(^)e-.^^i>(AA))
Tr ,(e-^^'i'(^')+^)
and therefore
In
particular
co^, l^^AA' |v)i^=
^A 0
^
Ö>A'
COA'\A
COA-
Next consider the infinite System and
Y,
W,^(\}=
assume
^(^}=
that the interaction energy
^o(A;AO
lim
A'uA?^0 yVuA^=i 2
norm. This energy represents the inter remainder of the System in the complement A with the action of the Subsystem A^ of A. The above observations then motivate the following definition.
is well defined, i.e.,
Definition 6.2.16.
the limit exists in
Let O be
an
interaction of
a
quantum spin System such that
the surface energy ^(t)(A) is a well-defined element of ^ for all A c L. A state a; over ^ is defined to satisfy the Gibbs condition with respect to if the
following conditions
(1)
co
is
are
faithful, i.e., QÖ, is separating for Tico(^]
CD^^
(2)
:=:
CO^^O)
for all A c L, where OJA is the local Gibbs state is
state
a
orem
All limit
over
^AS o}^^ is the perturbation of
5.4.4, and PA
=
corresponding co
points of the local Gibbs states satisfy the Gibbs combining Proposition 6.2.15 and the following
converges
strongly
to an
lim A^
for allA^^
and
O,
d)
condition. This result.
Proposition 6.2.17. Let ^ be an interaction of a quantum spin System the corresponding local automorphism group. Assume that T"^
to
constructed in The
^^CD(A).
follows from
(1)
ß^
fulfilled.
t G
automorphism
group T, i.e.,
||Tf(^)-T,(^)||=0
cxo
IR.
and T^
264
Models of Quantum Statistical Mechanics
The
(2)
surface energies ^$(A)
well-defined elements of
are
^
for
all
\CL. It
follows that to ß^, for
PROOF.
If
(T, ß)-KMS
every
all
respect
ß
state
satisfies the Gibbs condition
with
e U.
0, the implication is trivial. H ß ^ 0, it suffices, by rescaling, to ^=^-1. First let Q -(Wc^(\) + H^(\)) and consider the of the local groups T^ for all A^ D A. It follows from perturbation T^ assumption (1) and the definition of T^ that T^''Ö converges strongly to an automorphism
consider
^
=
the
case
=
'-
group
T^' which
^(i)(A),
acts one
trivially
on
^A and which leaves ^A^ invariant. Similarly if
P
=
finds
T''
=
T'^T'^"
,
we identify T'^' with its restriction to 2I/i'. But the corresponding perturbation ca'' of O) is a T''-KMS state by Theorem 5.4.4.
where
Next for 0 < 5 e
StA' define
the state
p,^, "^(^)
As T^
=
T^
(g)
But coA is the
over
äU by
w''(AB) =
^;/W
T^' the T^-KMS condition for o)^ implies that c/j^ is unique such state by Example 5.3.31. Therefore, co^
co^(ÄB) for all ^ E
co^
^IA and
0 < 5 e
=
a
t'^-KMS
=
state.
COA and
CD^(A)(D^(B]
^IAS i.e., CO^
=
COA
<^<^^kl^.
Thus the Gibbs condition is valid.
The
of this
converse
conditions
on
ditions
stated in
are
assumptions
which
Theorem 6.2.18.
corresponding (1)
proposition
the interaction >.
is valid under
We first derive
a
slightly
more
striogent
result in which these
con-
implicit form and subsequently give alternative explicit also imply the converse. an
Let ^ be
local
an
interaction
automorphism
T^ converges strongly
to
of a quantiim spin System andi^
the
group. Assume that:
an
automorphism
group i, i.e.,
litn||Tf(^)-T,(^)||=0 A>
(2)
OD
for allAe^ and t G U. The surface energies W^(A.)
are
well-defined
AC.L,
(3)
D={J^, \CL
is
a core
for
the generator d
of i.
elements
of
^
for
all
265
Quantum Spin Systems Then the follomng conditions
(1) (2) Remark. an
each
equivalent for
are
the Gibbs condition with respect
(D
satisfies
CD
is
a
(T, ß)-KMS state.
If ^ is
a
Symmetrie
derivation defined
interaetion O' sueh that d
=
on
to
ß^.
one ean always find energies ^^'(A) are well-
D, then
and the surfaee
ö^',
^R\
ß
defined elements of ^. This follows from the Remark before Theorem 6.2.4. The surfaee ever.
The
energies
same
defining (5, howprineiples in
not well defined for all interactions
are
remark
applies
to later theorems on variational
this seetion. If
PROOF.
j5
0, both conditions state that co is a trace. Thus by rescaling it suffices l. Moreover, throughout the proof we identify na}(^) and ß
=
to consider the case
=
^.
(2). By assumption, CD is separating. Let er denote the corresponding automorphism group. Now for each P P* G 51 the perturbed state co^ is also separating and has modular automorphism group cr^ where cr^ is the perturbation of <j introduced in Proposition 5.4.1. Choosing P= -^o(A), the Gibbs condition implies that o/ co/^^ co, where a)A(= CD^^) is the local Gibbs state on ^A and 60 is a state on 5t A^ As co^ is separating, it follows that both COA and cö must be separating. But the modular group of WA is the group T^ and we let f denote the
(1)
=>
modular
=
modular group of cö. Now of cD\cb satisfies the modular ((T, 1)-KMS) con T^ 0 f and therefore by the uniqueness of the modular dition with respect to T =
=
group, Theorem
5.3.10,
one
has 0^
T^
=
Next let (5^ denote the generator of
given by D(d^p]
=
G
Z)((5ff).
D(da]
Thus
particular
ö^lgj^
=
^^'
It follows that the generator
d^p of a^ is
and
d,(A] for all A
0 f. In
er.
d,p(A]-i[P,A\
=
0^ and T^ coincide
äs
^TA
on
concludes that ^CA C
one
D(df^)
and
d,(A]
=
i[H<^(\],A]-{-i[W^(\],A]
=
i
Yl
lim
[^W.4-
^A'W
^-"
XnA^0
for all A G ^IA? where
we
have introduced
D and
ö/^> by D(ö^>) ,
öAA}=iY,[^(X),A]
,
xcA'
and the limit exists in the
norm sense.
But if ö is the generator of T, it follows from Theorem 3.1.28 that ö is the
limit of
öj^'
äs
A'
-^
00,
hence A ^
D(ö)
Ö(A} for all A e D. But D is
a core
=
Ö,(A)
for ö. Hence D is
^
a core
for da and ö
Proposition 3.2.52. It follows that T: a, i.e., (1). This follows from Proposition 6.2.17.
3.1.19 and
(2)
=
If we combine this theorem with Theorem 6.2.4
involves the interaetion in
a
mueh
graph
and
more
explieit
we
co
is
a
obtain
manner.
=
Ö^by
Theorem
(T, -1)-KMS
a
state.
Statement which
266
Models of
Corollary satisfying
Quantum Statistical Mechanics
6.2.19.
Let
iii>iL
be
O
=
an
interaction
E^'"(^"p \'^^^
/2>0
for some 6.2.4).
(1) (2)
PROOF.
co
satisfies
co
is
a
are
aiitomorphism
The restriction
||)||^
6.2.4. Hence the two
< +00
is sufficient to
be derived from the
Suppose
(see Theorem
ß^.
State.
hypotheses of Theorem conditions are equivalent.
can
group
equivalent:
defined. The other two
EXAMPLE 6.2.20.
J
the Gibbs condition with respect to
(i^.ßYKMS
A similar result
qiiantum spin System
a
E ii'Wii)<+~ X3X \X\^n-V\
1 > 0 and let i^ be the associated
The follo\ving conditions
of
L is
ensure
assumptions
homogeneous,
that
W^(!^]
is well
6.2.18 then follow from Theorem
of Theorem 6.2.6.
the interaction ^ is of finite ränge
and
sup||(D(X)||
<
+00
.
XCL
Then the
co
State
to
assumptions of Corollary 6.2.19 are satisfied and a state if, and only if, it satisfies the Gibbs condition with respect
6.2.3. The Maximum The Gibbs states of
is
a
(T^, ^)-KMS
^.
Entropy Principle
isolated, i.e., closed, Systems
can
be characterized
by
a
of maximum entropy at fixed energy. This principle was outlined in Section 5.3.1 prior to Theorem 5.3.15 and it is often referred to äs the principle
principle
of minimum free energy. Our intention is to give a more detailed description of this principle for quantum spin Systems and to extend it to open Systems. The notion of entropy first arose in thermodynamics äs a measure of the heat absorbed, or emitted, when external work is done on a In the
System.
subsequent development of classical Statistical mechanics this quantity was related to the order, or disorder, of the microscopic particles which constitute the System. Thus, the principle of maximum entropy describes equilibrium äs the State of maximum disorder compatible with a given energy or particle density, etc. The concept of entropy was then abstracted from Statistical me chanics and used in both probability theory and information theory. In the former it constitutes a measure of the "uncertainty" and in the latter it measures the information content although a direct justification of these latter interpretations is difficult. Nevertheless, the entropy arises naturally in the
Quantum Spin Systems formalism of the law of large numbers and it is to its
Notes and
general Utility (see
apparently
positive
measure
M\(X)
with respect
=
if ^ is absolutely continuous with respect to v, and Thus if X is discrete with a finite number of
^(ft)/)
ii e
-y'rfv(a))(^g)(a>)log *)() -<'-(S))
5(/i|v)
-:
this fact which leads
Remarks).
S(^\v] of a probability measure v G Mj^(X] is usually defined by
The relative entropy to a
267
CXD
if this is not the
points
coi,...,co,
case.
and
=^/, then
A-,V(CO/)
n
^(A^IV)
=
-X^(A-iogA-- A'iog^f)
.
i=i
particular the entropy, S(fi), of the measure jj, which gives unit weight to each point is given by
In
with respect to the
measure
n
S(f^)
=
-Y^pi\ogpi
.
/=!
In the
sequel
we
need to
generahze
both these concepts, the entropy S(^) and spin Systems, and for the
to the states of quantum
the relative entropy S(j^\v), discussion of open Subsystems it is also necessary to introduce a third concept, the conditional entropy. We begin by recalling the relevant description of a
closed
Subsystem.
Each State
(matrix) subalgebra 9lA of the quantum spin unique density matrix p^ on ^^ in the form
of the local
CD
System is determined
by
a
0}(A} =TTf,^(p^A} for all A G ^A, and the entropy
S/,(o}) If
is
{A}I
a
complete
S/^(co)
of
co
is defined
by
-TrJpAlogpA)
=
set of
eigenvalues of
p^, this definition takes the
classical form n
S\(o})
=
-Y^pilogpi
.
/=!
(The
function
x
log
x
is
0.) If (p always defined by continuity to be zero at jc on ^A? then (p is also determined by a positive =
is any other positive functional trace class operator CTA, i.e.,
(p(A) =Tr5jaA^) and
one can
define the entropy
5(co|(p)
=
ofco
relative
to (p
by
-TrJpAlogPA-PAlogö-A)
.
268
Models of
Quantum Statistical Mechanics
Thus if PJ^ and CTA commute, n
S((D\(p)
-^(pi\ogpi
=
pi\og qi)
-
,
/=!
where {qi}\
demonstrates that this
Lemma 6.2.21. n
is
negativity
equivalent
to the
at fixed energy, for the Gibbs state of
entropy,
Let A be
a
positive
principle
matrix and B
n x n
of maximum
^Aa
strictly positive
n x
matrix.
It foUo\vs that
-^r(A log A with
and
equality if,
only if,
A
=
-
A
log B)
<
Tr(5
-
A)
B.
PROOF. with
Let {öt/}i<^< and {^/}i
-Tr(^log^-^log5)
=
-^a,log,/
=
+
l
^^a,|(.Af,^f)|2logZ.y i=\
.
j=\
But
5;]|OAf,iAf)|'iogZ>,-
<
iog^|(^,iAy)l%-iog(^,5^)
7-1
7=1
by convexity of jc
^
log
Therefore,
x.
-Tr(.,o.. -.,..,
<_
p,;. .*,',{- -^,0,-,^}
^S*'^'-w:s*?)}^"'<"-^'' where the second
inequality
follows from the -^
log
.T
<
l
convexity -
^
of .T
-^
-x
log
x
in the form
.
This
gives the desired inequality. But we have used two estimates and equality occurs in the first if, and only if, there is a relabelingy j(i) such that I/A<^ is proportional to i/^J( -) and equality occurs in the second if, and only if, a/ Z?y(/) Hence equality occurs in the total estimate if, and only if, ^ B. -^
=
.
=
Proposition
6.2.22.
Let
co
and cp be states
over
the local
siibalgebra ^A ofthe
quantum spin System with corresponding density matrices p^ and (j\. It follows that the relative entropy S\((D (p) is negative, i.e.,
S^(co\(p} and
S\(a}\(p)
=
0
if,
and
=
-TT^^(p^logp^-p^\oga^}
only if,
oj
=
cp.
<0
269
Quantum Spin Systems Moreover
if co^
is the Gibbs state
of ^A corresponding
to
the interaction O,
then
5A((ü|ü)*)
=
5A(w)
S/^(CD)
where the entropy
co^
is the
-
S^(cD]
-
PROOF. =
.
co(//o(A))
given by
logTrgJe-'^'C^)) B
,
which maximizes the functional
state
Ff,(cD]
^
and the maximal value is
-TTf^^(p^\OBp^)
=
unique
^2iA
e
co
logTrgJe-^*'^))
+
given by
is
SA(o}) Consequently
CD(H<,(^))
-
.
The first Statement follows from Lemma 6.2.21 with the choice A
=
p^ and
a/^ because
Tr(^-5)
=
Trs,(pA-<^A)
The other Statements follow from the first
termined
by
the
l-l=0
=
by noting
-
that the Gibbs state is de-
density matrix Q-H<,(^]
^^^Tr^,(e-^^(A)) Remark. The function
(D^E^^^ o}(H^(A]) is often called
energy of the
ihefree
the Gibbs state
co^^ is the unique
-
ß-^S^(cD) at inverse
spin System
temperature ß. Thus,
state which minimizes this energy. The term
free energy is also used for the minimum value
FA(^O)
=
-r^QgTr^J^-^^-^^))
of this function. In the lattice gas sponds to the pressure.
interpretation
In Order to extend the maximum entropy necessary to analyze properties of the entropy finite subsets A C L and of the state co G E<^^.
notion of relative entropy to by analyzing the entropy.
First, remark that
positive. Next,
^A, then iS'A(co|v)
<
[0, 1]
äs jc G
note that if
0
pairs
Moreover,
of states of the total
i-^
-
x
log
x
implies
is
1)1^1}
.
Thus 0 <
Sf,(cD)
<
corre-
one
must extend the
spin System.
We
begin
positive, the entropy must be (normalized) trace-state over
that
^A(co)
quantity
principle to open Systems it is S\(o^] both äs a function of the
is chosen to be the
v
this latter
|A|log(A^+l)
.
270
Models of
Quantum Statistical Mechanics
0 if, and only if, co Moreover, S(co v) v and hence S\(o}) attains its upper bound if, and only if, co is the trace-state. It should be emphasized that S^(o}) is an extensive quantity, i.e., it is es sen tially proportional to |A|, and hence the total entropy =
=
lim A
of
a
State
global
is
co
>
S/^(co)
oo
only finite under exceptional circumstances.
measurement of
..
hm A^oo
of
co
next
and
we
A
more
suitable
entropy is the entropy per unit volume
will examine this
quantity
S^((D) '/ |A| for
translationally
invariant states in the
subsection.
EXAMPLE 6.2.23. matrix
Let
determining CD\<^
be
co
,
product state over ^, i.e., if P{^| denotes the density density matrix p^ which determines co sij is given
a
then the
by
PA-n%) .vA
It follows
immediately
that
^AH
^5|.,}(CO)
-
.
xeA
Hence A
-^
S\(a})
is
monotonically increasing and lim
S/^(co)
=
only if, co|s)i sense
for
that
(D
one
eigenvalue
iS'{;^}(to) of
^
5'{Y}(cü)
is small if, and
p|^} is dose
to
one
is summable. But
only if, co|<)j
and the others
6'{^}(co)
0 if, and pure" in the dose to zero. Thus, =
is "almost are
have finite entropy the restrictions co must be predominantly "almost ^ij Thus, finite entropy is only possible for states which are pure in this very
to
pure." strong
sense.
Probably ditivity,
the most
Proposition ^A scitisfies
6.2.24.
all
important property
The entropy A c L
the strong
^A, for
finite if jc
only
is pure and
Y]5'{.^|(co)
f^L
^-^
is well defined but it is
the total entropy
u
of A
\-^
^A(^)
h->
Sj^(a))
G (R
of
is strong sub-ad-
ecich state
co
over
siibadditivity inequality
AZ uAs
(^}
-
SA^
u
mutiially disjoint AI AI, AS ,
^2
(^}
^ *^Ai
u
A3
(<^)
"
'^A, (<^)
c Z/.
The proof of this proposition consists of a rather long, but elementary, series of arguments based upon convexity and positivity or convexity combined with complex analysis. As this proof is not very illuminating we omit the details
271
Quantum Spin Systems
(see
Notes and
and
Remarks)
satisfy
ourselves with comments
on
two
special
cases.
S0(w)
Note that
=
0
0 in setting AI subadditivity property
definition and hence
by
subadditivity inequality
finds the usual
one
==
the strong
^AzuAsH <S^,((D]-^S^,((D] for A2
n
0. This explains the terminology strong subadditivity. Moreordinary subadditivity follows directly from Lemma 6.2.21 by
A3
the
over,
=
choosing A PA, AS and B^ p^^ p^^. Strong subadditivity is also easy to derive whenever mute. For example, if co is faithful, the p^ are invertible =
^
(^AiuA2uA3(ö;) -^AiuA2M)
-
(*5'A, uA3(co) =
the various p^ and
com-
iS'A, (co))
-
-Tr^^^^^^^^(pA,uA2uA3log^)
,
where ^
But
of
convexity
;c
l =
jc
l
PA]uA2uA3PA,PA,uA2/^AiuA3
log
gives
x
-log^
jc
<
log
^-^
jc
<
'
l
Hence
jc.
-H
and strong subadditivity follows immediately. Nonfaithful co can be handled by approximation. The next proposition summarizes various properties of the function A, co 1-^ *S'A(CO). Some of these properties will only be relevant in the following subsection when we discuss translationally invariant states and interactions. 6.2.25.
Proposition
For each A c L the function 0}
is continuous, when
5A(AC01 Für t her it
5AacOi
+
all
S/i((D)
G R
+
(l
^)C02)
-
the
>
A5A(c^l)
+
(l
-
).]S^(CD2)
concave,
i.e.,
.
"convexity" relation
(l-A)c02)
A^A(COI)
Hh
(l
-
A)^A(co2)
-
AlogA
-
(l
-
;0 log(l
-
;.)
triangle inequality
I^A, (co) for
\-^
E^ is equipped with the weak* topology, and
satisfies
<
and the
eE^
-
disjoint AI A2 ,
^A2 (co) l
<
5'A, uA2 (co)
<
^A, (co)
-f-
SA, (co)
C L.
If coa <^ in the weak* topology on Ey[, then co colgj^ in the weak* j^ topology on Ey^^^. But 51 A is a matrix algebra and hence C0a|^^ ^^^A in the uniform topology. In particular, the density matrices p^ corresponding to co oj^ converge in norm to p^. Thus, if {}Jf} and {A/} denote the eigenvalues of p^ and PA, arranged in PROOF.
^
>
~^
272
Models of
decreasing that
}Jf
Quantum Statistical Mechanics
order and
repeated according
multiplicity, perturbation theory implies
to
/i/. Therefore
-^
^A(CÜ)
=
-Y^^i
log ^^-^'^ A, log l,
/
S^(w)
=
,
/
i.e., S^ is continuous. To prove
and C02 in ^A' f'or 0)1
p^^^
concavity let
p;^^
and
denote the
density
matrices
corresponding
=
^
density
to
^A. Let E\ and 2 be two mutually orthogonal rank-one projections some A' with A n A' 0 and define co äs the state over Ul^ with
over
A'
matrix
p
Ap^^^^E,^(l-l)p^-^E2
=
.
One has
5AH-^A(AtOi+(l-;>2) and
'5'AuA'(^) Thus
concavity
-
^A'(W)
follows from
-
^S^(cOi]
+
(l
-
/1)5A(C02)
.
subadditivity.
Next note that
;.p<,')
+
(i-Ä)pJ,-'>Ap<,">o
and hence
iog(ip;,'^ (This
+
(i-;Op\-0>iogH"
last conclusion follows from the
log for A, B
>
0 and
^
-
log
5
=
.
representation
/co
'
'
-^
A +
n)
d).
\B + n
Proposition 2.2.13.) Therefore
-lTr^,(pj,') log(Api')
+
(l
-
;0pi'^))
<
-m^Jpi^^ log Api'^)
=
-Alogl
+
15A(<:üi)
.
Similarly
-(l
-
l)Tr,3jp(c^ log(/p^^)
+
(l
-
A)p|e^))
<-(i-;Oiog(i-x)
+
(i-;o^A(co2)
.
The
"convexity" relation follows by addition. right-hand side of the triangle inequality is subadditivity which results from Proposition 6.2.24. To establish the left-hand side we proceed äs follows. Let p be a density matrix on the /?-dimensional with eigenvalues space {A/},
m-dimensional space with an orthonormal basis {cp/}i
il^E r.
=
^-^'
2^^^/''^h^(pi /=!
it follows that
,
where
m
>
n
If E is
273
Quantum Spin Systems
Tr^^^.(^(^01))^Tr^(p^)
,
Tr^^^.(^(105))-Tr^KP^^)
,
p' denotes the density matrix on ' whose first n eigenvalues are corresponding eigenvectors (p^, and whose remaining eigenvalues are zero. where
Thus it follows from these observations that if
find A' with A
A'
co
is
a
^IA, then
state over
//, with
one can
(2) the corresponding to cö is one-dimensional, i.e., cö is a pure state, and (3) 5^(c5) S^(o}) Sj^'(a)). By a similar argument it follows that if CD is any pure state on ^AuA' ^A Sj^>(co). ^AS then S^(ü) Therefore, replacing A by AuA^ we can consider co, over ^I^uAS to be the restriction of a pure state cö over ^AuA'uA" f^r some A'^ C L, It follows from purity that ^ ^^d '^A'uA"!^) '^AuA'uA"(^) S^(^) =S/i(a}). Finally an application of Pro co gives A'', and CD A', A2 A, AS position 6.2.24 with the identifications AI
density
n
0 and
=
a
state
co over
such that
"S^AU A'
(1)
cö|gj^
=
oj,
matrix
=
=
-
=
=
=
=
=
-^AuA'H<^AH-^A'(^) Thus
of A and A'
by interchange
one
concludes that
^AuA'(^)>l'^AH-^A'HI
ing
is
6.2.24
Proposition
^AuA'(<^) ~*^A'(<^)
basically Proposition
^^^
this difference which
Proposition
we
concerning the difference gives further information concern the following.
Statement
a
6.2.25
summarize in
For A C L consider the
6.2.26.
(A', co)
G
A^
X
^51
^
^A,A'(^)
functional
^
^AuA'(^)
-
^A'(^)
It follows that:
E<^fixed, A'
A"^
(1)
For
(2)
|5A,A(aj)|
(3)
For A and
/\! fixed
the weak*
topology. A.' fixe d CD^-^ Sj^ A'(^)
(4)
CD
G
<
For A and
5A.A'(Aco, and
S
+
atisfies
5A,A'(ACO,
+
5A(co)
(l
-
the
(l
-
C
<
^~^S\^M(^]
decreasing.
|A|log(;V+l).
CD^-^
A)co2)
S^ A'(^)
>
<
^^ continuous
ifE^
is
equipped
with
^^ concave, i.e.,
A5A,A'(a;i)
"convexity"
;0(Ü2)
^^
+
(l
-
A)5A,A'(2)
relation
A5A,A'{CÜ1)
+
(l
-
A)5A,A'(W2)
-Aiog;.-(i-;oiog{i-A) for PROOF.
(2) (3)
(1)
0
<
A
<
This is
a
l and coi^coj ^ ^^21reformulation of
Proposition 6.2.24. triangle inequality of Proposition 6.2.25. from the continuity of 5'A(co) given in Proposition
This follows from the
Again
this follows
6.2.25.
274
Models of
(4) and
for
Let coi and co2, be states
p^l^j^'some
density
Quantum Statistical Mechanics
Next let E\ and 7 be
A^' with A'' n (Au A')
^^AUA' with corresponding density matrices p)^^^ mutually orthogonal rank-one projections in il^" 0. Now define
co äs
the state
over
^t^^A'uA"
with
matrix p
lp^^l^.^E,
=
+
(l-l)p^^l^,^E,
.
Thus
^AuA'l^)
-
SA'(O^}
^A,A'(>^1
+
(l
^S^^^'(cDi)
+
(l
=
-
^)C^2)
and
^^AuA'uA^Ö^)
-
^A'uA"(^)
=
-
^.)S^^^'(C02)
Concavity of ^/^ A/ now follows from Proposition 6.2.24. The convexity relation for S/^^^' is a consequence of the concavity convexity relation for ^'A, established in Proposition 6.2.25. The difference on
Sj^^j^'(o})
-
S^'(co)
measures
of ^A, and the
the entropy of the
spin System
AuA',
measures
in the state CD, minus the entropy of the System on A^ Thus, it the entropy of the A-subsystem in interaction with the A'-subsystem.
Therefore,
the entropy of the A-subsystem in interaction with the remainder of the lattice System is given by the limit of this difference äs A' ^ A"". The limit exists because A.' \-^Sj^^j^'(co) is decreasing by Proposition 6.2.26 (1) and this limit is often called the conditional entropy. It represents the entropy of the A-
subsystem
conditioned
by interaction
It is also referred to
distinction to
S^(co}
with the exterior.
the entropy of the open A-subsystem in contrawhich measures the entropy of the isolated, or closed, äs
Subsystem. Definition 6.2.27.
quantum spin
tropy
S\(CD)
Let
S\(a})
denote the entropy of the Subsystem A of the co G E^^. The corresponding conditional en
System in the state
is defined
by
5A(cü)=
inf
(5A,A,()-5A'(co))
A'cA' =
lim,(5A^A'H-5A'())
.
A^A'
The conditional entropy S\(co) is again an extensive quantity, i.e., essentially proportional to |A|. In fact, if co is a product state, then S\(co) S\(co) and we have already argued that S/^(co} is extensive (see Example 6.2.23). Several basic properties of *SA(CO) follow immediately from Propositions 6.2.24-6.2.26. =
Proposition 6.2.28.
The conditional entropy
S\(CD)
has the
follomng
erties:
(1)
For
(a)
G E^fixed, the function (Strong superadditivity)
co
A C
*5'AiuA2uA3(<^) -^AiuA2(<^) for AI, A2,
and
^
L^-^S^(co) *5'A,uA3(<^)
AS mutually disjoint.
G
~
D^
satisfies
'^Ai (^)
prop
Quantum Spin Systems
(^)
|^A(co)|<^A(a;)
For A. C L
(2)
continuous
A5A(cOi) >
fixed, the function co if E<^ is equipped with +
(l
^)^A(C02)
-
Sl^(X(Di
+
(l
-
e
}.]0}2)
.
E
the weak*
Alog A
-
>
275
-
(l
^S/,(o}i)
G [R is upper semi-
topology
-
+
A) log(l
(l
-
and -
;0
1)S^(0}2}
0 < l < l and coi , C02 ^ E^-
for
PROOF.
(l a)
By definition
S'A, uA2 uA3 (<^) =
AO
>
+
S'A,
uA,
(ö^)
-
^'A, uAj (
lim ^{(^AO u AI u A. u A3 (AiuA2uA3)
+
M
^'A, (to)
-
'^'AoH)
('S'AouAiuA2uA3(<^)
'5'AouA3(<^))
(^AouAiuA2uA3(co)
'^AouA2uA3(<^))}
=
AO
-
-^
lim ^{-6'AouA2uA3M (AI uA2uA3)'^
('^AouAiuA2uA3 (<^)
'^AouA2 (<^))
+^AouA3(^) +'5'AouA2M -^AoMI
>o
where the
inequality
(Ib)
This is
immediate consequence of
(2)
By definition ^'A
these
are
an
follows from the strong subadditivity Q^ S\((ß]. (2) of Proposition 6.2.26.
is the infimum over the family of functions 5/^^A' '^A But by Proposition 6.2.26 (3) and hence ^A is upper semi-conconcavity-convexity relations follow directly from (4) of Proposition ~
continuous
tinuous. The 6.2.26.
Proposition 6.2.22 established that the Gibbs states of a closed Subsystem given by maximizing the entropy at fixed energy. But the conditional entropy measures the entropy of an open Subsystem of the infinite System and our aim is to demonstrate that the states satisfying the Gibbs condition are exactly are
those which maximize the conditional entropy with the conditional energy By conditional energy we now mean the energy of the open Subsystem,
fixed.
i.e., the Subsystem in interaction with its surroundings. This principle of maximal conditional entropy is also referred to äs local thermodynamic stability. If one refers to Proposition 6.2.22 one sees that the maximum entropy
principle
for closed Systems follows from the
negativity
of the relative entropy
of finite Systems. The same is basically true for the infinite spin System but in Order to establish this it is first necessary to generalize the notion of relative
entropy. We such that
pairs
next extend this notion to
QCÜ
is
separating
for
n(j^(^)"
of faithful normal states of
discuss the
corresponding
pairs
a von
.
of faithful states of ^, i.e., states
In fact
co
define relative entropy for Neumann algebra and then use this to
notion for states of
we
a
C*-algebra. (It is possible to directly
introduce the relative entropy of nonfaithful states but this is not relevant to the discussion of equihbrium states.)
276
Models of
Let 9[R be
^i5 ^2
a
Quantum Statistical Mechanics Neumann
von
algebra acting on a Hubert space are both cyclic and separating for closure of the operator defined on 9}J^2 by
be two vectors which
^
reintroduce
S^^^^,
the
äs
5^^,^^2=^*^l This operator was already discussed in 2.5.33 established closability. Let C, ^c^l,C2 .
denote the
polar decomposition
o^
.
2 and in
Chapter A
and let
9Jl. Next
particular
Lemma
V2
/ ~^Cl,6^Ci,C2 -
,
S^^^^^.
The
selfadjoint operator
^^,,C2=^C*C2^^i'C2 is called the relative modular operator of (9JI, c^i, following, useful, definition of relative entropy. Definition 6.2.29.
(^2)-
It is instrumental in the
Let coi and co2 be two faithful normal
positive linear funcand and QI 301, algebra ^2 the unique vector representatives of coi and co2 in the natural positive cone ^ ^^^ ^^2- The relative entropy of coi and C02 is defined by tionals
over
a
Neumann
von
=
=
/oo
l
S(a)i\co2}=
(^(Qi,^(l)Qi)logi
,
Jo
where E is the
spectral family of
the relative modular operator
/oo dE(^)X
AQ,,!^,, i.e.,
.
_
Let CD[ and 602 be two
positive forms
over
normal extensions a)\ and cü2 to 9Jl relative entropy of coi and co2 is then defined
unique
=
C* -algebra such that their
a
(na^^
© 71^02
(^))''
are
faithful. The
by
S(oji\cD2} ^S(a)i\ä)2) We
now
analyze
First note that
various basic
Ay^^^
yj^^^
^(^iCOi 1^2^02) and in
=
=
of the relative entropy. (l2/^\}^Ü2,ü, and hence
properties
Al^(cOi|oJ2)+/lllog(/i2/>^l)a)i(l)
,
particular
S(10}1\10}2)
=
1S(COI\C02)
i.e., (co l, 0)2)^-^ S ((Dl 0)2) is homogeneous of order Next remark that
log /i
<
1
-
,
one.
l and hence
/"OO
^(coi|a;2)<
/
JQ -
J(Oi,^(A)Qi)(i- 1)
(Oi, Ao,,n,Qi)
-
(QI,OL)
=
(02,^2)
-
(ni,Qi)
.
Thus if coi and co2 are states, (Q2,^2) l (^1,^1), and one concludes that *S'(coi|a;2) is negative. Moreover, S(co[\(D2} =0 if, and only if, E(X) is con=
=
Quantum Spin Systems centrated at the
S(coi\a)2}
point
1=1. But this
if, and only if,
0
coj
=
only
occurs
if
An,, DI
=
277
H and hence
co2-
In contrast to the entropy and conditional entropy the relative entropy of States of a spin System is a global quantity. Nevertheless, one can argue that S(a}i 0)2) takes finite values only for pairs of states which are globally similar.
This
qualitative notion will be illustrated in Example 6.2.31. It motivates the following examination of the entropy of a state CD relative to a perturbed state
Cü^. Let
(D
be
faithful state
a
cyclic and separating perturbed vector Q^,
on
and the
cbP(A} äs
the
Neumann
algebra 9JI with corresponding selfadjoint P G 9Jl we can define the corresponding perturbed positive form von
vector Q. For each
(Q^,AQ.^}
=
Ae'ü
,
.
in Theorem 5.4.4. Note that co^ is not normalized and cb^
co^ is the perturbed
state
given
in Theorem 5.4.4. Now let
=
us
\\Q^\\~co^, where
recall
some
of the
basic facts derived in Section 5.4.1. First
one
has
Q''-r^,y2Q, r^f/2 is the analytic continuation to the point i/2 of the unitary cocycle which intertwines the modular group er associated with (501, Q) and the perturbed modular group cr^ which corresponds to (501, Q^). If AQ denotes the modular operator corresponding to (501, Q.) and H log AQ is the infinitesimal where
=
generator of
er, then
F!./2 Now
F^./2
-
^(^+^)/2^-^/2
assuming for simplicity that G 9Jl. Therefore, S^
But SRQ is
a core
F^^/2*50Zß
is
r^//2*^^
=
^*r^,-/2^
for both SQ_ and
a core
P is
.
analytic
=
^*^^
with respect to
er
one
has
V, n^^
=
S^p Q. Moreover, F^^y2 immediately that
is invertible and hence
for iS^. It follows
Sa
r_//2*
=
^^^^,0
Consequently
AQ^Q
=
r!^-/2Aar!,//
=
(g(^+^)/2^-^/2)g^(e-^/2e(^+^)/^)
=
e(^+^)
and
logA^F^ These relations then extend to
=
general
logAQ-fP P
=
.
P* G 9J1
by approximation in the One uses, for example
strong topology with sequences of analytic elements P. ||Q^" Q^ll ^ 0. Thus we have deduced that -
278
Models of
Quantum Statistical Mechanics
5(cüCüO
=
(^,logAj2P_nfi)
=
(Q,logAnQ)
where the last Step follows because AQÜ
(Q,PQ)
+
Q.
=
=
cü(P)
By rescaling
also obtains the
one
identity
S(cD\CO^)=CD(P)-log\\ü^\\But it follows from
interchanging
co
Corollary
and co^
one
(co^)"^
5.4.5 that
.
co
=
and
(O^)~^
=
O. Thus
finds
^(co^|a;)
-co^(P)+log||0^||'
=
Combination of these two relations then
provides
.
lower boimd for the
a
re
lative entropy. One has 0 >
Thus both
S(cD co^)
S(co\co^)
S(cD\(D^)+S(co^ co)
>
and
S(a}^\a})
expression yields
a
S(a}\CD^)
>
-2||P||
.
[0, -2||P||]. combined with the
the bound
of the
generalization
of
nonpositivity
iiQ^ir This is
c/(P)
-
take values in the interval
It is of interest to remark that the
above
(ß(P)
-
>
e"(^^
inequality
Tr(e^+^)
=
/Tr(g^P)|
^"""^{T^^^j
-
Tr(e^) which is valid for matrices //
.
/f * and P
=
P* and is known
the Peierls-
äs
Bogolitibov inequality. (One can also derive the upper bound co(e^) > ||O^|r corresponding to the Golden-Thompson inequality Tr(e^e^) > Tr(e^+^)). These relations and bounds will be of importance in the sequel. EXAMPLE 6.2.30.
Let coi be
a
faithful state
over
the
algebra 9JI,,
of
/?
x /?
matrices
and pi the density matrix determining coi. Furthermore, let co2 be a positive faithful form determined by the positive matrix P2- Faithfulness implies that pj and p2 are
invertible and hence
one can
define H\ and //2
Hi
=
\ogpi
by
.
-H2-Hi. Thi
S(w\a}^)
=
co(P}
gives
S(coi\co2) Thus the
general
nition for matrix
in
coi(//2 -H\]
=
-Tr(p, logpi
-
pi
Iogp2)
.
definition of the relative entropy coincides with the
previous
defi-
algebras.
EXAMPLE 6.2.31. äs
=
Let coi and co2 be
Example 6.2.23,
and
product states over the quantum spin algebra, they are both faithful. The representations tensor äs products of the representations
assume
(fo,, ^oj,, ^w,) can be realized (Sfo,,5 ^fu;,5 ^oj,x) associated with
co/^.^
=
co/
sji
.
The relative modular operator is
Quantum Spin Systems
expressible äs consequently
a
S(co,\co2)
product
tensor
of the
^5(,,,|co2,,)
=
corresponding one-point
In this last
jc
p^'^
expression G Z/
1^
denotes the
of p^^^ and
-^
because
=
-Tr5,,,(p(')(logp(')
sufficiently fast
p^^\
matrix which determines co/,.x and the 6.2.30. Thus for finite relative entropy the Z^ this means that '^ 0 must be summable. Now if L
density
Example
S(coi^x\(^2,x) xer
must vanish
.
xeL
identification follows from function
relative operators and
^-Trö,(pi')(logp(') -logpf ))
=
xeL
279
|jc|
äs
)^,^j
-^
logpf)))
-
<
But this function is
oo.
0 a
continuous function
has dimension A^ + l, and it vanishes
Thus for finite relative entropy implies ||coi,;c <^2,A:|| ^ 0 äs
one
|jc|
must have >
||p^^^
p^^^ ||
0
-^
only if p^^^
äs
;c|
-^
i.e., the restrictions coi,.x and co2,x
CXD,
=
p^^^.
But this
CXD.
must be
asymptotically equal. If coi is Z^-invariant it is
sufficient to
that
ensure
possible
S(a}i\co2)
to estimate the rate of decrease of
is finite.
/
/*
log.-log5 for
=
Again l
||p^^)
pP||
the representation
we use
l
| ^;{^^-:^^
matrices A and B to deduce that
positive
|^(coi,;c|a)2,x)|
pi')/ rfA(pf (roo
=
+
Ai)-'(p(')-pf)(p(')
+
;j)-'
/O
s/- rfA|i(pi"+Air'iiii(P<'^+^^ir'iiiiPi"-pf ^0
=
r ^^{(11 pF'-'
r' +^r'
-
(II
Jo
pi'^-' r' +^0}
p(')-pp
p<"-' II-' =
-
II
pf '-' II-'
iogfMJ:;i)(ii p(')-' r' \ll
P'-c
II pi^)
-
Therefore, ||
pi^MH p^^^~^ ||
0 -^
äs
||
PQ
-
r')-' ii P!" -PP ii
II pf-'
II/
.
|:c|->oo and || pi^^-MH II Po^^"Ml by II äs |:c| oo, and we conclude that
|5(co,M|<||p^^^-MlEllAl^^-pf Next
we
entropy of
a
derive state
an CD
extension of the rule before
relative to the
Proposition 6.2.32. algebra 9K and for
perturbed
Let coi and 0)2 be
each
selfadjoint
D.2
with CO2'
^^ ^^^
II
=
.
Example
6.2.30 for the
state co^.
faithful
states over
P G 9Jl introduce the
&^,(A} where
ZMnvariance.
-^
the
von
Neumann
perturbed form
(Q^,^Q^)
perturbation of the cyclic and separating
vector
associated
280
Models of
Itfollows
Quantum Statistical Mechanics
that
S((ß\\cb^) The
PROOF.
proof
S((D\ 0)2)
^
^
03\(P)
.
combines the argument used to derive the relation
^(coi a)f)-coi(P) and the 2x2 matrix arguments used in Section 2.5.4.
Adopt the notation of Lemma 2.5.33, ^\ n\\ ~^ ^i^nr^ by an operator of the P^E22
By
Lemma 2.5.34
one
=
and
the vector QQ
perturb
=
^ol'^h ^2)
form
^21^^21
U22PU22
+
has that 7
logA^,
Y, ^log(A.^^)t;^
=
/,y=i
and hence
^0(^1,^2)^^"
-
=
But if A
Oirt,Qo)
=
exp H and J
^(^^S^"o+^^-)/2^-^ogA^^/2^^(^^^^^^^) Ci (g)^ii + C2 ^fe
the modular operator and
are
(*)
no(Ci,c2)
=
conjugation
associated with
then
AF
=
exp{//
is the modular operator associated with
Corollary 5.4.5.)
Also
by
Lemma 2.5.34
+
P'
-
JP'J]
(9[R, QQ'). (A proof one
of this appears
prior
to
has
J=Y,U,jJ,^,,,U], and hence
J(PE22]J It
=
follows, using P' =PE22.
log A^,,(.,,.p)
equation
we
equality
is
-
log A^^(,.,,,.^)
P (g)
=
logA^(^.^)
+
a
+
U,,PU^_,
U^^_J^_^^.^PJ.^^.^Ul^_
consequence of
this to Ci
=
QI and ^2
S(cOi\a)2)
we
-
+
U22PU^
Uj_^_J^^PJ.^U'^2
(*). Taking
the
^
(2, l) component of this
obtain
applying
Next
.
"22
logA^.p^^.^ =logAc,,,., and
U^^^PJ^JJ^
that
-
where the first
t/l2^c,^,,C2^^
=
=
^2
one
+P
deduces
S(COI C02) -^CDi(P}
.
examine the connection between the relative entropy and the
conditional entropy.
Quantum Spin Systems
281
Let coi and coi be states over the quantum spin algebra and {PA JAci ^^^ ^^^ families of A' matrices. If c corresponding density A*^, then JAci
{/^A
^AuA'(coi)
^A'(coi)
-
Tr(p^^l^. logpi^^A') + Tr(pj,V logp^V) ^-Tr(pi^i^,(logp(,^)^,-log00pj,^;))) + Tr(pj^V(logp^V-logp^';)) =
=
-
|a)2|^ .) ^(<^ik, ,,KA(8)a)22t .)-S(a;i|3i AuA' "=ctA" v
^A'^
where
,
have used the identification of the relative entropy established in
we
Example
'-'^"A'
6.2.30 and TA denotes the
(unnormalized)
TA(^)=Tr,(^) This relation
can
be translated to
a
trace
^G^A
,
on
^A, i-e.,
.
relation between the conditional and
A^. entropies by taking the limit A' continuity properties of the relative entropy. lative
^
But this
re
requires knowledge
of
Proposition 6.2.33. Let ^ be a von Neumann algebra with an increasing net of finite-dimensional subalgebras 971 such t hat U ^a ^^ dense in 50l. Let 0)1 C02 be two faithful, normal, positive, linear functionals over 93^ and ,
^i,a,<^2,a. their restrictions to SR. It follows that ai-^'S'(coi^a|<^2,a) is
monotonically decreasing
\imS((D\^^\(ß2,a}
=
and
S((D\\(I>2]
a
Although this is a key result, the proof is rather long and so we choose to (see Notes and Remarks). Combination of this result and the discussion preceding it gives a relation between the conditional entropy and the relative omit it
entropy which is fundamental for the derivation of the maximum entropy
principle from
Corollary
the Gibbs condition.
6.2.34.
Let coi and 0)2 be states
and let TA denote the It follows that
(unnormalized)
S^(o}i} =5'(cOi|TA(8)a>29i^J This follows from the relation limit A'
^
over
trace over
-
the quantum
spin algebra 51
^A-
5(601 |5J^JC02|5I^J
preceding Proposition
6.2.33
-
by taking
the
A".
Proposition 6.2.35. Let ^ be an that the surface energy W^(A.) is Furthermore, let
at
be
a
state
interaction a
which
of a quantum spin System such well-defined element of^for all A CL. satisfies the Gibbs condition mth respect
to O.
It follows that
S^(CD]
-
ßo)(H^(A}}
=
sup
{^A(coO
-
ßo}' (ff^(\}}}
Models of
282
for
Quantum Statistical Mechanics
each A C L, vvhere S/^ is the conditional entropy,
Ät>(A)
=
H^(^)
+
W(^(A.)
the conditional energy, and
C^
{co'; CD'
=
G
E^j, CD'
co
=
,^^^
.
The Gibbs condition states that
PROOF.
CO/'^MA)=^^|^^^_^^
,
where COA is the finite-volume Gibbs state. This is
^ßH..W where
.j^^ J
T^
is the
(normalized)
^
trace state
Proposition
to
^0 ^A, i-C-,
on
<(^)=1:M' It follows from
equivalent
6.2.32 and
'^^^-
rescahng
that
5(
condition, Definition 6.2.16, and Corollary 6.2.34 give
S(co,|co''('^))
5(cO||
=
5(<:oi|TAco|.,,J-logTrö,(1l)
=
5(cü,|u^^, co|,^J+5A(cü,)-logTrö^(1l)
=
these relations
Combining
S/,(cDi )
=
one
setting
0)1=01
5^(0;) and
this
-
S(cDi 1,1^^ \oj
.t^^^,
)
+
ßcoi (ÄD(A))
log{(nf''(^),Qf''(^))Tr^Jl)}
.
gives
/?o,(//
=
.
finds
S(o}i |co) +
Now
.
+
log{(nf*<'^', a/f""^')Tr5,(1)}
hence, by subtraction,
^AH
^co(//ci>(A))
-
Therefore, if
coi e
^AH
-
^A(COI)
-
^co,(^ci>(A)) -^(co,|co) +6'(coi|,i^^Jco|s,i^^J
Q^ -
^CO(ÄD(A))
-
~S^(cDi}
-
^COI(ÄD(A))
>^A(coi)-jÖcoi(Ä,(A)) where the
inequality
is achieved if
Note that
co
=
.
follows from the
coi, the
S(O}I\CD) C^^ for
=
0
proof
is
negativity complete.
if, and only if,
co^
-
S(WI\CD)
,
of the relative entropy. As
=
co
and hence the state
equality
co
is the
state in
which the supremum is attained. Next we establish a converse of Proposition 6.2.35 under stronger assumptions on the interaction. In fact, we use the assumptions that sufficed to
unique
establish the
eqiiivalence
of the Gibbs condition and the t-KMS
condition.
283
Quantum Spin Systems Theorem 6.2.18. This is
because the Gibbs condition
slightly unsatisfactory
and the maximum entropy principle can both be defined when the surface energies W^(A.) make sense and it is not essential that the dynamics are de-
strongly continuous one-parameter group of *-autoniorphisms T hypothesis in the following theorem appears redundant. (See Remark after Theorem 6.2.18.)
termined
by
a
of ^. Thus, this also the
Let ^ be
Theorem 6.2.36.
local
corresponding (1)
T^
an
strongly
(2)
automorphism
to an
group T, i.e.,
||Tf(^)-T,(^)||=0
lim A>
and T^ the
group. Assume that
automorphism
converges
of a quantum spin System
interaction
cx)
for allAe^.te U. The surface energies W^(A)
well-defined
are
elements
of
^
for
all
Acl.
(^)
I^-U^A ACI
is
a core
the gener ator d
for
The following conditions
(1)
SA(a))
-
are
of 1 G [R and for
equivalent for jS ,
ßo,(ff^(\}}
sup
-
{^A(coO
-
a
state co G
"91.
ßo,\H^(A})}
co'eq
for +
all A C L, where S/^ is the conditional entropy, W^(A) the conditional energy, and
C^ (2)
PROOF.
follows bound
co
(1)
is
{co'; 0}'
(T, ß)-KMS
a
=^
=
(2)
lfß
:
=
by choosing T^ given in Proposition co'
=
6
^21, 0}'
^^^
=0)^^^}
*S'A(CO)
If
^ 7^ 0,
0 and
0
co
^^^
T^
.
denotes the
(normalized) trace state on ^A, it principle and also invoking the
in the maximum
6.2.28 that
achieves its maximum value the
H^(A.}
state.
^A(T^)<5A(COO<5A(CO) Thus
ff^(A.}
^A('Z^A)-
.
Hence
proof is
cojgj^
=
T^
and
co
indirect. We argue that condition (1) implies that the autocorrelation lower bound and then appeal to Theorem 5.3.15.
is
a
co
satisfies
trace.
First note that if ö denotes the generator of T, then
Ö(A) for all ^ G ^IA
defined
by
the argument
=
given
i[H^(A),A] in the
proof
by
w,(B)=co(e^^^Be-^^^)
of Theorem 6.2.18. But if co^ is
284
Models of
for all 5 G
Quantum Statistical Mechanics
2t, then condition (1) implies that
^{co,(Äp(A)) whenever ^
=
A* e
-
co(//ci)(A))}
>
^A(CO,)
~S^(co)
-
^A. But it follows from the defmition of SA and properties of the
trace that
SA(cot) and
=
S/^(co)
consequently
^{co,(Äi.(A)) Thus,
t^^
at this
ßcot(H,i,(A))
has
-
minimum
a
CO(^CD(A))}
at /
.
0 and hence its first derivative vanishes
=
point. Therefore
ßco(ö(A)) for all A
e^A and A
choose 5
G
B'AB
=
-
{B^'B.A}/!
^U, it follows that
pointwise invariant
yß(A]^^ for semigroup r > OH^ T/
under the
^*^kuc. We
0
ß ^ () this shows that co is invariant under Ö. y^;^^^^!, introduced prior to Theorem 5.3.15, i.e.,
yB(A) we
=
and since
c L
Next consider the operator
If
0
>
already argued (1) that
=
^
.
all A
G
2l A^.
exp{?7^},
=
Therefore, ^IA^-
is
and hence
yL^c
in Section 5.3 that T*E^n C E^n,
t >
0. Thus it follows from
condition
^{(7;*oj)(Äi>(A)) Next
by
the invariance of
-ißco(B''d(B)]
-
CD
CO(ÄD(A))}
under ö
=
-iß{(D(B''d(B]]
-
=
-iß{cn(B-^ö(B))
-
one
>
^A(r
-
^A(CO)
.
has
co(ö(B-')B)
+
co(ö(B''B))}/2
CD(d(R^)B]}/2
ßw(ys(ff^(A}))
=
.
Thus, the previous estimate gives
-ißcD(B*ö(B))
=lim
ß{(T*co)(H^(A))
-
CO(ÄD(A))}/^
>^lim_^{^A(7;*co)-^AH}A Next note that
äs
r*co
=
i^,
^l^i^,
one
5v(r for all A' C A''.
=
.
has
5^,(w)
Therefore,
5A(r-5A(co)= iim(5A,A,(r;.o)-5A,A,H) A -^A'^ But if A''
=
AuA',
then
5^^^(r-VH--
[^ds^T:r^^(p\Aogp\.)
Jo where
^j^".
p^/,
denotes the
But each
p^,/
.
is
a
,
""^
density matrix which determines the restriction of T*(o)) positive invertible matrix, and hence the expansion
to
285
Quantum Spin Systems
/.oc ^00
logp>
=
JO
//
l
^M^ ^.l Y
+ U
A +
p^,
is valid. It follows that />OO
J
/
J
l
l
-Wlogp^)=y^ .A-Tr(p^^^^_^^ =
r.^Tr(^f^-^ l^i V'"'
7o
1
A + p^ J5 /l +
p^J
C/p"
,
'^
-Tc/ATrf'^^V we
have used the the
'
ly
cyclicity
of the trace and
dropped
the
subscript A^'.
expansion roo
D
-
/
7o one
"l
'
^
^'^'l,^l,in"iT7'^'^(lT77;
'70 Using
^' + p-
+ 11
l
l
where
^^_^^^
l
d^p^
(A
+
pO
2
then derives
^Tr(p^-log/) Tr(^^(logp^ ^(r(log/
1)
+
=
y;(r(log/
=
+
(r(ya(logp^
=
1)U,
+
=
1)
+
1))
=
(r(7slogp^)
.
Therefore,
S^..(T:OJ)-S^(O})
=
f'ds(T:o,)(ys(logp^^.,))
-
.
Jo
The next step is to bound the
Let
integrand. n
PA"
E^V^>
=
7=1
denote the
spectral decomposition of
p^". A
straightforward
calculation shows that
n
-co(75(logpv'))
=
X^
S(;>.j;^^)Cjt
j,k=\
where
Cj,=J,^^((E,BEj}(EkBEjr) and the function
jointly
convex
^^(w
;
v]
=
^ S(lj ;4)c;i >S[Y^ 9*^V J,k=\
is defined
u\og(u/v)
in its two arguments and also
\J,k=\
;
homogeneous of first
E '^^^^M J,k=\
in Theorem 5.3.15. Now S is
äs
)
=
S((a(B'B)
;
order. Therefore,
(55*))
.
286
iS"
Models of Quantum Statistical Mechanics
Finally combination gives
of all these estimates
-ißo}(B''ö(B)}>
/
lim ^-^o +
ds
together with lower semi-continuity of
S((T^a})(B^B) ](T'^co)(BB'^})/t
Jo
S(co(B*B) ;co(55*))
=
for all 5 G ^A. But the union of the ^A is
a core for ö and hence this inequality semi-continuity to all BED(ö), i.e., the auto-correlation lower bound is valid. Condition (2) then follows from Theorem 5.3.15. (2) =^ (1) : If ^ 0, then co is the unique trace-state on ^. Thus co is a product State and S/^(a}) Sj^(cü) achieves its maximum value. If ß ^ 0, then the implication follows by combination of Propositions 6.2 AI and 6.2.35.
extends
lower
by
=
=
6.2.37.
Corollary satisfying
Let
be
<^
an
interaction
of
a
E ^'"P^P E II^WII)
11^11;^=
n
V''^^
>0
^^^
qiiantum spin
System
<+^
/
\X\=n+\
for some /l > 0 and let T* be the associated automorphism groiip (see Theorem 6.2.4). The follomng conditions are equivalent for each ^ G K: (1) (2) (3)
the Gibbs condition with respect
co
satisfies
co
is
0}
satisfies
the maximum entropy
S^(cD]
ßo3(H^(^])
a
(i^,ß)-KMS
-
to
ß^,
State,
=
sup
principle,
{5A(coO
-
ßo3'(H^(\)}}
a;'eC-
for
fl// A c //, given in Theorem 6.2.36.
This is sures
elaboration of
an
that the
Theorem
hypotheses
Corollary
6.2.19. The restriction
of Theorems 6.2.18 and 6.2.36
||)||^ are
< +00 en-
fulfilled
(see
6.2.4).
Note that if I is
Section 6.2.1, then
interactions
homogeneous lattice,
a
Corollary
has
äs
defined at the
beginning
of
concept of finite ränge interactions. For such 6.2.37 establishes the equivalence of the Gibbs condi
one
a
tion, the KMS condition and the maximum entropy principle whenever
SUp;^Cl||^WII 6.2.4.
<00.
Translationally
Homogeneity discussion of
Invariant States
of the quantum
spin System
unimportant in the foregoing physical Systems exhibit worthwhile analyzing such features was
equiUbrium spatial homogeneity and symmetry and it is in more detail. Throughout this section we assume
states. Nevertheless many
that L
T and examine Z^'-
287
Quantum Spin Systems
Z^' is not of paraparticular choice L could be modified to subsequent arguments importance cover Systems associated with other periodic sublattices of IR^. Some of the results are also vaHd for appropriate non-invariant interactions or even for aperiodic lattices. We return to a discussion of these points at the end of the section. We derive results of two types. First, we establish a global variant of the maximum entropy principle for Z^'-invariant states and, second, we deduce some general Statements concerning uniqueness and lack of uniqueness of equihbrium states. invariant states and interactions. The
=
and most of the
mount
E^
denote the states invariant under the action T of the group Z^' of it translations of the quantum spin System on the lattice Z^'. If co G space follows that the local entropy S/^((D) has the invariance properties Let
E^,
S\+a(o^] =^A(CO) for all and
Z^, and A
G
a
we
Z^. But
c
aim to examine its
is also extensive, i.e.,
^A(CO)
mean
value
äs
proportional
to
tence of the mean it appears necessary to restrict the manner in which A
If
edges
fl
=
of
(l,
.
.
.
length
flv)
,
Z^ and üi
G
>
0,
we
define A^
{jc;
=
z
The translates of A^
by
tition ^a of Z^. Let
n'^(a)
Z', 0
G
vectors
na
< jc/ < fl/,
and
rx(fl)
tend to
the
=
(nia\,
n~^ (a) denote the number of
.
.
z
.
=
l,
.
,n^,av}
.
.
,
v}
co.
with
with
.
n
r^(fl)
sets of
Z^' form
G
partition
in the
of
sense
van
a
parwhich
be the union of these sets.
^a which
are
contained in
the union of these latter sets. The finite subsets A c Z^'
infinity
>
parallelepiped
denote the number of sets of this
have nonvoid intersection with A and let let
äs
l:
/
A
Similarly
|A|,
A-^oo. In order to establish the exis-
are
A,
defined to
Hove if
limiM
=
,
n^(a) for every partition ^a and in this case we write A-->oo. Note that if A (a) denotes the set of points which are contained in A and a
distance greater than
|A|
>
\a
=
^
|A(fl)|
>
supi<^
(X W
-
from the
(A()
-
of
boundary
X W)2')|A.|
are
A, then
.
Therefore
i>4^>i-f>-5SV2'+i)^ |A| V <(), -
Thus
äs
A-->oo
one
has
shows that the notion of
|A(fl)|/|A| van
-^
1. A
slight
elaboration of this argument oo and equivalent to |A|
Hove convergence is
IAWI/IAI-I. Proposition 6.2.38. Let A c Z'h->5'A(co) entropy of the J.^ -invariant state CD.
G
[0, |A|log(A^+ 1)]
denote
the
288
Models of
Quantum Statistical Mechanics
It follows that the limit
5()
lim
=
A>oo
^^(") |A|
exists and
N
e/ S(a)]
The functional
PROOF.
G
co
E^
continuous vvhen
"'
is
^(co)
-^
^A.H ;\ IA^I '
-r mf az;
=
.
.
[0, log(N +1)]
G
mth the vveak*
equipped
/^ o^/^e topology.
and iipper semi-
The
proof relies heavily on the subadditivity and strong subadditivity Proposition 6.2.24. First, define S by
AH^IS'A(CO)
established in
5(co) Next let A~
A\rx(a),
=
|A;|
inf "sr,
=
.
then
|r+(a) -rx(a)|
<
%^ |Aa
=
(+(a) -xW)|A.I
Consequently,
^A;H But
äs
<
K(a) -^XW)|Aa|log(7V+ 1)
is subadditive and /^-invariant
Ah^5A(co)
one
.
also has
^AM<^rx(a)H+%H X (Ä.H
<
KW
+
-
n-^(a))\A,\\og(N
Therefore,
^^<M^+f^iM_iVog(^+i) UxW / ^^^^ |A| |A| +
>
-
Similarly,
if
A^
r^(a)\A, subadditivity gives
=
SA(O})
>
5'r+(a)M-^A+(w)
.
Hence
^AH^:^r>)H > |A| Thus for each
g
>
-
0
one
/
'
^A^"^^
.
_
|r;tWI
n-^(a) l _:v\v:v
may choose
5A.() IA.I
4Wy
V
<
a
n ioafA^4. ^^ ^
such that
S(w)
+
E
and it follows that
SA((O) ^
/5>)M
^M-^WniSr-' |A| V inWI -
for
sufficiently large
A. Next
we
bound the
right-hand
side.
+
1)
.
of
289
Quantum Spin Systems Let
US
ordering"^
construct
F^ (a) by successively adding translates of A^
of the vectors
the Union of the first
n
n
=
(i
.
,
.
.
,
assume
lexicographic denote
Similarly, if Z? m^a^) with m/ > 0, then (mia\ by successive addition, in lexicographic order, of translates of translates.
==
,
Ab can be constructed Afl. Let A denote the union of the first Now
in the
which define these translates. Let r
v)
n
.
.
.
,
of these translates.
that
Sr,,(cD)-Sr(co}<(S((D)-B)\Aa\ for
some n.
The strong
subadditivity of S\ immediately implies
5A,, H
S^^(CO)
-
<
(S(CO)
-
that
s)\Aa\
into A^n+i with the property that a translation of r+i r+i is mapped onto the last translate of A^ in A;+i This is the case if the last translate in A;;,+i is not "too near" the surface of A/,. For the remaining m subadditivity gives
for all
m
such that there exists
the last translate of A^ in
.
5A.,,(co)-5Aja;)<|A,|log(7V+l) Therefore, if b is large enough,
5A.()
=
.
obtain
we
^(5A(cü)
S^^(w))
-
(S(ca)
<
-
e/2)\Ak\
m
which contradicts the definition of
iS'(co). Thus,
5rH-Sr(co)>5(co)|A| for all
n.
Hence
^rj(a)()
=
E^^r,,, ()
-
Sr,(co))
>
S((o)\r+(a)\
.
n
Combining
this with the
previous
estimates
we
have
S(cD)+8>^^>S(co)-8 for
sufficiently large A. This establishes the first two Statements of the proposition. afi[inity of CD H-^ S(o}) follows directly from the concavity and convexity relations given in Proposition 6.2.25. The upper semi-continuity follows because S is the infimum of the functions S\^((o}/ |Afl| which are continuous by Proposition 6.2.25. The
Remark. and has
Proposition 6.2.28 establishes that S\((D) concavity and convexity properties similar
identical argument
one
N
'^AH
r hm
=
A-oo
is
a ^
an
affine function
over
'^'
Gibbs state with respect to This
ordering
is defined
strongly
subadditive
S\(oj). Thus, by
an
deduces that
e/ S(o})
defines
is to
by setting
n
,' |A|
^A^(co) =
sup
ez;
with values in some
< m
;\ IA^I
'
,
[S(co},S(co)]. Moreover,
if co
interaction O for which the surface
if nj < nij where j
=
min{z;
/
/ TW,-}.
290
Models of
Quantum Statistical Mechanics
energies l^o(A),AcI.,
|A|
-^
then
00
S(co)
are
well defined and
S(co).
=
This is
a
ßo}(H^(^))
>
satisfy ||PF(D(A)||/ |A| corollary of Proposition 6.2.35.
0
^
äs
One
has
~S^(CD) for CD'
co|^2iA
=
^
^
-
^^^
"ii^c-
S\(co')
=
^A(coO
S/^(co)
ßcD\H^(A))
-
and hence
^AH-2||FFc,(A)|| <^A(CO)<^A(CO) Therefore
S(co)
entropy and the States
"!{
The
(see
=
S(o}). Despite
identity
on
the Gibbs states the
conditional entropy do not coincide Notes and Remarks). mean
and
new
this
.
somewhat
feature
surprising
of
on
the
mean
the invariant
mean
entropy
'!' \^S(co}
is the affine property. This reüects an The entropy of a mixed phase is obtained
6
empirical physical phenomenon. by superposition of the entropies of the pure phases in the appropriate proportions. In the sequel we will give a more detailed account of this interpretation. The affine property also has an interesting corollary for periodic states. Let co be a state which is invariant under the subgroup CD
Zl where b G and
one
Z^.
=
The states
{ba
(biai,...,b,a,}',a^r}
=
{T*CO; a
e
A^}
are
periodic
under the
same
subgroup
verifies that
^Ac5)-^A+a(co) Hence the
mean
entropies
are
CD
=
S(a))
is the Z^-invariant state obtained
(o(A}
.
equal,
^d)) Thus if
,
=
.
by averaging
|A,|-' ^ c5(T_,(^))
A^, i.e.,
o) over
,
fl6A/;
the
affinity
of S
gives
S((a)=sl |Ai|-' ^ V This
equality
Next
we
will be of
examine the
state. This is
basically
weaker conditions
infinitesimal
on
T>
aeAft
=
|A,|-' ^ 5(T =5()
.
aeAfi
/
subsequent Utility. mean
energy of
an
invariant interaction in
an
invariant
closed System problem and can be handled under the interaction than those used to construct even the a
dynamics (see Proposition 6.2.3.).
Proposition
6.2.39.
Let ^ be
System and siippose that
a
Z.^' -invariant interaction
of
a
qiiantiim spin
Quantum Spin Systems
i
5^e2a<,o^
=
\^\
X3(^
Further let
CD
be
a
291
Z.^ -invariant state
over
^.
that the limit
Itfollows
//^(co)
a)(//a>(A))
hm
-
j |A|
A-->oo
exists and
H^((D) where E^ ^ ^ is
O}(E^)
=
,
defined by
\-^(X}
F
^*=Ä^WThus
E'^\-^H^(co)
G
0}
is
affine,
an
\H^((o)\ PROOF.
weak* -continuous
<
\m
function satisfying
.
One has
co(//a.(A)) |A|
-
w(E^)
|Ar'ü)(VA-^ (DW ^^T.i'a, )
=
-
C A
=
A:
-|Ar'E E -^'^.At',..
But
,
given
>
e
0
one
may choose
a
A
/
(O.) i;,, ' '
such that
vMM<,
^
W
'
D(X)>a
where
/)(^)
is the diameter of X, Hence
CO(//CD(A)) |A| where
A(a)
from
the
-
o^(E^]
<^+(^)wi-
indicates the set of
points in A which are at a distance less than a already remarked that if A oo, then 0. Hence H<^((D] exists and is equal to (ß(E<^]. It is evident that H<^ is |A(üf)|/lA| affine, weak*-continuous, and the bound follows because \\E^\\ < ||O||. now
boundary.
But
we
have
~->
-^
Remark.
If
the energy of the open Subsystem, in the above discussion then it appears necessary to ^o(A) H^(J^) ^a)(A), conditions . for on If, place stronger example. =
one
+
replaces //o(A) by
292
Models of Quantum Statistical Mechanics
Y^mx}\\<+<^
,
X3Q
then
easily deduces
one
that
HmJSM^O |A|
A->cx)
and hence lim A
for each
co
G
>
a}(H^(A))=
lim A.-^
oo
o}(H,^(A)}
=
H^(o})
OG
E^
Propositions 6.2.38 energy functional
and 6.2.39 establish the existence of the
mean
"free-
' '
CD G
for
a
large class
E^^Fß^((ß)
=
S((D)
-
ßH^(co)
of invariant interactions 0. But in the
previous paragraph of equilibrium states were characterized äs the states which maximize the corresponding finite-volume function F^O,A(<^) S\(o}) ßcD(H^(A}}. Moreover, the maximum value of the function was directly related to the equilibrium free energy, FA(^<X>), this section the finite-volume
=
-
FAO)
=
-ß-^ logTr^,(^-^^-(^))
=
-ß-^
Fß^^^(o3)
sup
.
co^E^ii^ Thus
formally one expects the states which maximize F^(CD) to correspond to thermodynamic equihbrium states and the maximum value F($), of F^(co), to be proportional to the equilibrium free energy per unit volume. The remainder of this paragraph is devoted to the examination of this tentative interpretation. First we examine properties of the "free energy". invariant
Theorem 6.2.40. $ mth
Let S denote the Banach space
iw
=
thermodynamic free
^(0))=
lim --^
lAr'logTrsJe-^^C^))
(X)
exists for a// O G S and has the
(1)
1^ l
energy
A
follomng properties:
F(0)=
sup CD
^
where S and H^ denote the
(2) (3)
interactions
Eä^<+^ ^30
The
off -invariant
norm
{S(cD)
mean
(D G S
is
)
is continuous and
G
K^F(a)) 33 i-^F())
-
H^((D)}
Eil
convex.
entropy and
mean
energy
Quantum Spin Systems
|F(0)-F(^)| forall^,^
293
||
<
e^.
For simplicity we only establish that the limit exists for parallelepipeds A^ 1,2,..., v, all tend to infinity. The general convergence is edge lengths üi.i subsequently obtained by comparison arguments using the matrix inequality
PROOF.
whose
=
logTr(e^)-logTr(^)|<|M-5|| First, it follows from the last
Statement of
IAJ-^ logTr^,(e-^-(^))
>
.
6.2.22 that
Proposition
|Ar^(5A(co)
-
co(//o(A)))
.
Thus, if
(0))
lA^
lim inf
-
~^
(^-^^(A.))
logTr^^
^
a^oo
then
(0) for all
E^
G
0)
.
S(co)
>
H^(CD)
-
Consequently,
F($)
>
sup O)
Next consider
{S(o})-H^(w)}
.
4;'
of Z'' formed
partition
a
G
by
translates A
=
A^
+
na
of the
paral-
lelepiped A^. If A^
A=U^".
'
/=!
we
define
a
matrix p^
density
A by
on
((S)l^e-"^^^"^) (Trs,^(e-^<^(^.)))^
P^
=
.
These
density
under
Z^.
matrices determine
The state co^ defined
(a,(A)
periodic
a
state co^
over
^, i.e.,
a
state invariant
by =
|A,r' Yl ft).(T,(^)) JC
6
Afl
is then Z^-invariant. But
S(cD,)=S(ü,)
=
|A,rHcö,(//a,(A,))
equality follows from by explicit computation.
+
logTr,^(e-^*(^))
where the first
the discussion
the second
Now it is
and therefore
lim
lim
a^oo
b-^oc
given
e
S(o},) for all
a
> (jQ-
Thus
0
>
-
one
(Oa(H9(\h)) lAfcl
preceding Proposition readily established that
Wa(//*(A.)) |A.
=
0
may choose OQ such that
H^((0a)
>
,
lA^r' logTrg^^ (g-'/^lA"))
_
g
6.2.39 and
294
Models of
Quantum Statistical Mechanics
limsup|A^
"'
logTr^^^(e-^'"(^"))
< e+
(2)
The
F(^)
//cDH)
.
previously
obtained lower bound
gives the
and the Identification of Statement
convexity
F(m
-
we^J'
Combination of this estimate with the existence of
{^(co)
sup
"
a->oo
(l
+
of
F(^)
A)^)
-
follows from the
{S(a})
sup
-
6
co
(1). affinity of
A//o(w)
-
-
(l
> -
>
//{i)(co).
Thus
l)H^((ß)}
"1j'
{S(co)
H^(oj)}
-
to6'|''
(1 -^)
+
-.^F(
Bauer maximum
prindple,
upper semi-continuous function an
extreme
point
+
{S(co)-H^(co)}
sup
J'
^
W
(1-;.)F()
.
Lemma 4.1.12, estabUshes that each
compact
over a convex
of K. Thus, for each (^ e 93 there must exist
F(^]
S(CJO,^)
=.
-
//(D(CO(I))
convex
set K attains its maximum at an
cü^ such that
.
Therefore
F(O)
^
<
Interchanging
0 and T
S((ü(^]
F(^)
-
+
H^(o}^)
Remark.
F(^)
=
The
log(A^
//vp_(^(cO(i)) <
F(^)
+
11^
-
0||
.
gives
F(^)
+
//VP_,I,(COCD)
continuity
+
\\^-^\\
relation.
continuity relation can + 1) and therefore
be used to bound F, e.g., if T
'|F(0)-log(7V+l)|<||(D||
=
0, then
.
Next
above of
we examine properties of the states which maximize S AD. In the proofof continuity of Fwe already remarked that for each
maximizing
states
AO
=
{(D- F(^)
=
S((D)
-
H^(O})}
is nonempty.
Moreover, the discussion of the maximum entropy principle for and closed open Systems given in the previous paragraph strongly indicates that these states correspond to Z^-invariant thermodynamic equilibrium states.
Thus, it is natural criteria which
we
to ask whether these states
condition. The first and most general
Proposition 6.2.41. System
m
statisfy
the various
equilibrium
previously examined,
Let ^ be
a
e.g., the Gibbs condition, or the KMS result in this direction is the following.
J.^' -invariant interaction
satisfying
iM
=
i:^<+J^fBO
l"^l
of
a
qiiantum spin
Quantum Spin Systems
(2)
The
surface energies {^^(A);
Z^}
A C
liminf^
(3)
Itfollows
that each f -invariant to
O maximizes S
exist.
0.
|A|
A->oo
mth respect
=
295
state
which
satisfies
the Gibbs condition
H^.
Remarks
(1)
Assumption (1) assures similarly assumption (2)
that
co
G
E^^
\-^
H(^(a}}
e [R
is defined and
is necessary for the Gibbs condition to be defined. Thus, the essential new condition is the third. All three
assumptions
are
easily
verified if
^||(D(Jf)||<+^ X^O
and hence this condition could be taken
(2)
the sole
hypothesis. proposition satisfying the Gibbs condition. The existence problem will be solved subsequently under more stringent assumptions on ^ (see Theorem 6.2.42 and the discussion preceding it).
PROOF.
Let
{PA}ACZ^'
äs
does not establish the existence of invariant states
The
co
denote the Z^'-invariant state which satisfies the Gibbs condition and
t^^ associated
density
matrices. Moreover, let COA denote the local Gibbs
state, i.e.,
g-'f^ct(A)
coA(^)=Tr^^(aA^),
^A
Thus, the conditional entropy of col^j
S(^\<^^\ ^A)
=
-
;jr^^z^^
with respect to COA is
.
given by
-Tr^JpAlogpA -PA^ogfiA)
-5AH-a;(//a>(A))-logTr^(e-^-(^)) But the Gibbs condition
.
gives coA
and hence it follows from the
=
'^*W|,
monotonicity
of the conditional entropy,
Proposition
6.2.33, that
0>5(H.iJcoA)>5(co|co^^*(^))
.
Moreover
5Hco^*(^))>-2||^,>(A)|| by
the discussion
preceding Example
leads to the bounds
6.2.30. Combination of these observations
296
Models of
0 >
to this
Up
Quantum Statistical
5AH
-
co(//ci,(A))
Mechanics
logTr^J^-^'^(^))
-
>
mean
//o(co),
energy
and free energy
F(<^),
Next
maximizes S
=
Q
,
H<^.
examine the
we
-
lead to the relation
S(CO)-H^(CD}-F(^) co
-2||^a>(A)||
point Z^-invariance was irrelevant but now if one divides by |A| and takes oo, then assumption (3) and the existence of the mean entropy S (CD),
the limit A
i.e.,
>
States that maximize S
relationship
between invariant KM S states and the
H^.
The KMS condition has
an advantage over the Gibbs condition insofar äs deduce the existence of Z^'-invariant KMS states. If the interaction $ is such that T^ exists, i.e.,
one can
\\lf(A)
lim
-
^^^^cp(A)^^-/r//o(A)||
^
0
A^oo
for all^ G let
co
^,^
be such
a
G
[R, then
(i^,ß}-KMS
state and define cö
states exist
CD(A)=M(CD(I(Ä))) where M is
an
invariant
mean over
invariant. But it also satisfies the invariant. This follows
by Proposition
6.2.15. Now
by
f
.
(T^,^)-KMS
by observing that
,
It follows
T^
and
immediately
that cö is Z^'-
condition whenever is Z^-
if
commute
and hence
ö)(^Tf^(5))^M(co(T(^)T|(T(5)))) =
Thus cö is
a
Z^-invariant
M(a)(i:(B)i(Ä)))
(i^,ß)-KMS
=
Ö)(BA)
.
state.
Every i^-KMS säte co satisfies the Gibbs condition by Proposition 6.2.17. Hence, if > satisfies the supplementary hypotheses of Proposition 6.2.41, then co
will maximize S
The
//$. We
now
aim to derive
a converse
of this Statement.
of the
converse depends upon a number of approximation, conproof tinuity, and density arguments. Instead of trying to isolate a minimum number of implicit properties of > which ensure the validity of these arguments we specify an explicit space of interactions, the Banach space of Theorem 6.2.4.
Theorem 6.2.42.
Let ^ be
a
IIIL
=
f -invariant interaction
of
a
qiiantum spin
System such that
EII*WII^"'"< + X3(^
for some /i > 0, and let i^ denote the associated dynamical group of*-automorphisms of the spin algebra ^. If 0} is a 1.^ -invariant state and ^ G [R, then the following conditions are equivalent
297
Quantum Spin Systems
(1) (2) (3)
PROOF. orem
(i^,ß)-KMS
coisa
State.
satisfies the Gibbs condition with respect maximizes S ßH^.
co CD
First note that i^ exists and
UA c r' ^A
is
ß^.
to
for its generator
a core
by
The
6.2.4. Moreover,
ii^iio
=
E ii'^wii
< +~
X30
Hence, W(^(A) exists and
iAr^ii^o(A)ii
.
-^^;.n740 Therefore, the combined hypotheses of Corollary 6.2.19 and Proposition 6.2.41 are satisfied. Consequently, (1) <^ (2) =^ (3). It remains to prove (3) => (1). This will be achieved by use of various convexity properties. Let
||O||^
the Banach space of Z^ -invariant interactions It follows from Theorem 6.2.40 that the free energy
denote
B;[
< +00.
A-~
which
|ArMogTr^^(e-^*(^))
lim
F(^)=
for
00
exists for each ^ e B;^ and defines
a convex
continuous function
over
5;^. The key
observation is that each state coo) which maximizes S H^ defines a tangent functional to the graph of F at O. Convex analysis can then be fruitfully used in the examination of the
maximizing states. a convex, continuous, function
Recall that if J^ is element
is said to be
G X*
f^
tangent functional
a
F(x + ^)>F(x) for all
+
over a
to the
Banach space X, then of F at x if
an
graph
f,(^)
Moreover, there is
a unique tangent functional to the graph of F at ;>c if, ^eX. only if F is differentiable at x and in this case the tangent functional is the derivative dFx of F at x. For example, if F is differentiable at x and 0 < /z < l convexity gives
and
,
F(x + ^)
-
F(x)
>
Ä-' (F(x
+
h^)
-
F(x))-^dF,(^) h=0
and
dFx is
a
tangent functional
at
x.
But if
M^)
,
then
/.(^)
=
h-^f.m
<
h-^F((x + h^]
-
F(x))-.dF,(^} n
=
.
0
dFx. Replacing ^ by ^ in this bound one finds the opposite bound and hence fx Thus, differentiability implies that dFx is the unique tangent functional. The converse is easily deduced by arguing that nondifferentiability leads to nonuniqueness. =
-
Let
US
now
maximizing
reexamine the
state CD^. One has
relationship
between the free energy
F($)
and the
298
Models of
F()
Quantum
+
^)
Statistical Mechanics
>
S(a)^)
=
S(CO<.1))
//cD+4'(cocp)
-
//(D(CO(D)
-
//xp(cO(D)
where E^^ is the interaction energy at the
^XJ/
z:^ =
and
we
defines
-
origin,
\X\ principle together to the graph
tangent functional /^
a
CO(^(E^)
>
^
have used the maximum entropy
CO(D
F(^)
^Y(X) X30
Thus,
=
with
Proposition 6.2.39. through the
of F at 0
relation
/ci>(^) Now
we
begin
the derivation of the
=
-co^(E^)
(T^^, 1)-KMS
.
condition for co^
by examination
those )
for which CO(D is unique, i.e., those for which the graph of F at > differentiable. (The (T'^^)-KMS condition for coß<^ and ß e [R\{0} follows by
of is re-
scaling.) The free energy F is defined
the
by
thermodynamic limit
A
>
oo
of the
mean
free
energies
FA((I.)
=
and each of the functionals 0
|A|-'logTrs^(e-('^))
^Bx^^F^()
is
and continuous
convex
by
the
ar-
guments used to derive similar properties of F. The FA are, however, simply defined in terms of finite-dimensional matrices and it follows that they are everywhere dif ferentiable. Therefore, there is
unique tangent functional /A/D given by
a
derivative of FA at O, and this is
F\
to
at
^, the
,,,-|Tr,,(e-^W//>p(A)) ,^._ -iA| AoCP) .
-
Tr6A(e-^W)
Next for
AQ
C A and ^
J* G
^AO
one can
by setting ^(Ao +:c) T^(^) for all x e Z^ of AO. Adopting this definition one sees that =
A 6
^AO^COA,(D(^)
=
l
l
^
Z^-invariant interaction ^ 0 if A' is not a translate
a
T^(A')
-=
the linear functional
-f^,<^('y(A+A^]/2) ,
^
define
and
-
if\,<^('^(A-A^)/2i)
Tr^^(.-^-^(A),^(^))
^
Tr^^(.-^'^(A))
AO+.VCA
is both
positive
and normalized. Hence coA,(i) is
with any extension to the complete compactness of F^i that A c Z^' i-^ COA,
a
spin algebra o
state
over
^AO
which
we
identify
^. It then follows from weak*
has weak* -limit
points. Therefore, there exist
AO,^/A,,(D(^^)
which converge for all {^^;^ =^* G ^AÜ,AO C Z'}. But the linear span of the latter set is norm dense in 5;^ and hence A ^ /A^^O is weak* convergent. Each limit point /p is, however, a tangent functional to the graph of Fat nets
/o(T)
=
lim/A,,cD(^)
<
lim
a
(FA, (O
+
T)
-
FA, (0)))
-
F((D
+
T)
-
F(O)
.
299
Quantum Spin Systems Since
we are
Aa
/Aa,(i>
^-^
F to have
assuming
have the
a
topology. Moreover, combining the
C0(j)(^)
unique tangent
limit and hence A c Z^
same
O, it follows that all the
-^
/A^O
+
nets
converges in the weak* and /A, o), one condudes that
identifications of /$
0}^(E^(A+A^]I2)
^
at
i^^(E^(A-A^]l2i)
-M'^(A-^A^]I2] iM^(A-A^]l2i) lim (/A,(D('F(^+^.)/2) + if\,<^(^(A-A^)/2i)) A
=
-
=
-
^
cx)
Tr,,(.-^^(A)z.(^)) TrsJe-''(A))
Hn, IM-, y 2.. Acol^l
=
Are
Z''
AO+-XCA
for A G
^AO
2.nd all
AQ
C
Z^'. Thus, the
unique
state CD^ which maximizes S
H<^ is
limit of space averages of the local Gibbs states co^. It is this identification which allows the deduction of the (T^, 1)-KMS condition. the
thermodynamic Aß denote
Let
a
cubic array
AQ
=
{x]
Z^',
6
X
a
< x/ < fl, /
It follows from Theorem 6.2.4 that for each
e
>
l
=
0, Z?
,
.
.
.
,
v}
Z+,
e
.
one
may choose ÖQ such
that
||^/r//a,(A.)^^(^)^-.Wo(A.) A^ and all a F;J;^(0 defined by
for all t\-^
jc
e
> ÜQ.
/7^.^M_1A|-1 ^,a>W-|A|
satisfies the KMS
Tf 1^(^)11
Next remark that for
^,
<
5 E
^^AO
the
analytic
function
Tr,,^(^-^^'(-^)T,(^)g--^^^(^)T,.(^)e-^^^>W)
^
1^ AQ
_
Tr,,(.-^.(A))
+.rCA
identity
l\:l(t+'}=p^'^(-t} But from the
previous
estimate and the convergence of COA,^ to co^
\\m^F^^'^';,,(t)
=
w(ATf(B))
one
condudes that
.
Therefore, CO(D satisfies the (T^,1)-KMS condition äs a corollary of Proposition 5.3.12 (see the discussion following the proof of this proposition). It remains to handle the O G 5;. for which the tangent functional to F at <|) is not This is achieved by an approximation technique based upon the fact that
unique.
automatically differentiable at a "large" set of exceptional points a general tangent functional can be constructed points, by convex combinations of limits of unique functionals. For example, if Fis a convex function of one real variable it is automatically continuous, differentiable at all but a countable set of points, and at these points each tangent functional is a convex combination of the left derivative and the right derivative. The generalization that we need is contained in the following lemma. convex
continuous functions
are
and at the
Lemma 6.2.43.
Let F be
Banach space X and let z er o.
f
a
convex,
^ X"" be
a
function tangent functional to continuous
separable graph of F at
over
the
a
300
Models of
Quantum Statistical Mechanics
It
follows that f is contained in tangent functionals ^ defined by ^
=
{g GX*;
there exist x
F is
We omit the
>
differentiable
at
0
the weak*-closed
(in norm}
convex
hüll
of the
of
set
such that
each x^ and
linia dF(xy](x)
proof of this result (see Notes
and
=
g(x)^
x
X}.
Remarks).
END OF PROOF OF THEOREM 6.2.42. It follows from Lemma 6.2.43, and the weak* compactness of KMS states, that it suffices to prove that c^o satisfies the (T^^, 1)-KMS
condition in the
special
case
CO(E)
=
lim cücDa a
where
6
E^
have the
(1)
properties
ll^a-^||,-0
(2)
S
-
//(D^
is maximized
by
a
unique
,
state co^^
.
But it follows from this second property and the
satisfy the(T^", 1)-KMS condition. Moreover, the discussion following Example 6.2.5 that lim
uniformly condition
Finally
for
t
Ulf (^)-T^(^) 11=0
in any bounded interval.
by Proposition
foregoing analysis that co^^ must it follows from the first property and
Therefore,
it remains to prove (3) =^ (1) when ^ G; is a trace-state. We argue by
maximizes S, then cannot be
a
trace for
cocp satisfies the
(i"^, 1)-KMS
5.3.25.
sufficiently large
0, i.e., we negation. If
=
A and hence
co
must prove that if CD
^^^cannot
is not
a
trace,
co
co
sit
maximize S/^, i.e.,
S^(CD)< |A|log(7V+l) for A
sufficiently large.
But this
implies
S(a})
co
does not maximize S.
Remark. We have chosen to prove the equivalences of Theorem 6.2.42 under the assumption $ G 5;^. This can, however, be replaced by the hypotheses of
Theorem 6.2.6 and this is of interest for The identifications
one-
and two-dimensional Systems.
provided by Theorem 6.2.42 give a new criterion for equilibrium states, maximum mean entropy at fixed mean energy. The theorem is slightly unsatisfactory insofar äs the proof of equivalence of the three criteria is only established under relatively strong assumptions on O, i.e., O G 5^ for some A > 0. The Gibbs condition, and the maximum entropy pro perty, can of course be defined under much weaker hypotheses, e.g., ^ BQ, and it would be of interest to derive equivalence of these two conditions under invariant
301
Quantum Spin Systems
assumptions on O. This would demand a converse to Proposition particular a proof of existence of /^'-invariant states satisfying the Gibbs condition which is independent of the existence of dynamical group i^. Despite these shortcomings the theorem motivates the study of the set of states which maximize S H^ äs candidates for Z^'-invariant equilibrium states O. the interaction to corresponding
weaker
6.2.41 and in
Let ^ denote the Banach space
Theorem 6.2.44. actions O mth
norm
of J.'' -invariant spin
inter-
given by
=
vep<+ 1^1
^ X30
and
define Ä^) A*
C
E^
by
|(w;ft) E^,S(co)-H^(o}] e
=
-//o(o;')}|
{S((o')
sup
=
'^
...l
where S and //$ denote the
mean
,
J
cl^ ^'eEf; ^
entropy and
mean
energy.
It follows that:
(1) (2)
weak*-compact face of E^. Simplex mth ^(A,D) C S(E^] and the unique barycentric decomposition of each co G AO coincides mth the unique decomposition of (D into extremal I.''' -invariant states.
A(D A(D
/^
fl
is
nonempty,
a
PROOF
(1)
It follows
once
again
from the Bauer maximum
Lemma
principle,
4.1.12,
and the upper semi-continuity of .S //o that AO is nonempty and in fact Weak* AO must contain an extremal /^'-invariant state, i.e., an co
^(E^)-
compactness follows by noting that if
Iim{5(c0a) -//ci)(a))}
<
G
co
{5(co)
-
A(D and
Wa
-^
//
a
by
upper semi-continuity of 5" Similarly if we assume that
CO(D
with
co
1,602 G
"1*
but
=
}.<J}\
//o- Consequently, AO, and
co
G
AO-
coo G
+
coi^AcD,
(l
A)C02
then the
affinity
oi S
H(^ implies the
contradiction
sup
{5(co)-//ci)(co)} >/{5(coi)-//(D(coi)}
+
(l -/){5'(a;2)-//(D(co2)}
coeEl^ =
S(o}^)
-
//(D(cO(D)
=
sup 0)
Thus A(D is
a
face.
E
J'
{S((jo)
-
H^(a})}
.
302
Models of
(2)
Quantum Statistical Mechanics
The a
Spin algebra is Z^'-asymptotically abelian in the norm sense. Thus E^^ is Simplex by Corollary 4.3.1 1. The remaining Statements follow because AO
is
a
stable face of
Note that if A(D is
'|j
.
composed of
point
co, then
is
automatically an properties in mean discussed in Section 4.3.2. Interpreting the elements of A^ to be invariant equilibrium States, this indicates that if there exists a unique /^'-invariant equilibrium state, i.e., a unique Z^-invariant thermodynamic phase, this state, or phase, is au tomatically Z^-ergodic. Therefore if A(D consists of more than one state, the unique decomposition of each element co G AO into extremal elements corresponds to the Separation of the mixed thermodynamic phases into pure phases, i.e., Z^'-ergodic states. But the mean entropy S is affine and upper semione
co
extremal Z^-invariant state and it has the düster
continuous and therefore it respects the Corollary 4.1.18, e.g. if ^i,, G
barycentric decomposition
of
co
,
by
M^(E^^},
CD(A)
=
j d^l,,((D')(D'(A],
^ G ^
and
S(o,]
=
j d^i,,(cD'] S(o,']
.
Thus, accepting the foregoing interpretation of mixed phases and one
concludes that the
entropies.
entropy of
mean
a
A similar conclusion is valid for the
We conclude with various observations
mean
6.2.42.
phases, mean
energy.
concerning
the set of states A^
by elaboration of the convexity arguments Throughout these observations we take ^i to
used to prove Theo be a Banach space of
which follow rem
pure
mixture is the mixture of the
invariant interactions which contains the space So of invariant finite ränge interactions äs a norm dense subspace and moreover we assume that Si C S where S is the space of Theorem 6.2.44.
Observation
fiinctionals /$
1. to
There the
CO(D which maximize
is a one-to-one correspondence between tangent graph ofthefree energy,
/^(VF) This
was
=
almost estabhshed in the
-C0cl>(^vp),
T G
Sl
.
proofof Theorem 6.2.42. In
particular
we
showed that each state cod) which maximizes S //CD defines a tangent functional /(D through the above relation. Moreover, äs So C 25 1 each state is determined by its values on [E^^]^ G Si}. Therefore, distinct maximizing states lead to -
distinct tangent functionals and if there is a unique tangent functional /$, there is a unique maximizing state co^. But if /$ is not unique, then by Lemma 6.2.43 there exist G 25i such that
/CD(^)
=
fy^
is
unique, ||(I>a
lim/cD^(^)
=
(I)||
-limcoa),(^vp)
>
0, and .
303
Quantum Spin Systems But this
has the
implies that co^^ converges in the required correspondence with /$. The
Observation 2. norm
i
for
which
topology
A^
to a state coo which
consists
of a single point
is
dense.
This follows
By
<|) G
set
weak*
from Observation l and Lemma 6.2.43.
immediately
extension of the latter lemma
an
one can even
show that this set contains
countable intersection of dense open subsets. The implication of this last Observation is that the O for which A(D is a singleton forms a "large" set and in a
the
physical interpretation this implies that the spin System tends to have a pure thermodynamic phase "almost always". Nevertheless, it may happen for a fixed 0 that t^ß^ consists of more than one point for all ß in an interval of the form (ß^, 00 ) (see Section 6.2.6). It is, however, possible to prove the existence of a dense subset /) C Si, such that for O G D and Lebesgue-almost every j8 G IR the set A/^o is a singleton. Observation 3
4"
=
U
^*
OeSi
where the bar denotes the
If F is
a convex
uniform
closure.
continuous function
called F-bounded if there is
a c
G IR
over a
Banach space X, then
/
G Ji"*
is
such that
F(x)>f(x]^c for all jc G ^. One can prove that the tangent functionals to Fare the F-bounded functionals. NO W suppose
G
CD
Fjj
because
0 and
then
//o(co)
>
S((D]
co('o).
=
/(O.) is F-bounded
(with
bound
-
//o(co)
-03(E^]
>
Thus the function =
zero). Therefore, by
s by Observation
Moreover, if ^
=
^^(7) =0 if F is 11^^ II- 11^ |. Thus,
\M^]-f(^)\ ^~ ll^ll
the
_
l the tangent functionals
^* G not
^j^ and ^A is a
/
G
S^
defined
by
-a)(o)
there must exist
But
dense in
,
F(O) S >
norm
y^^
f^^
general to the
result just graph of F
cited at
^^
0
determine states coo.,
defined' by ^^(^ + .T)
-
G
A^).^.
T^(^),
and
translate of X, then the 33-norm of ^ satisfies
304
Models of
Quantum Statistical Mechanics
|ft;
|A(^^)-/(>I'^)| \\^A\\
and
consequently the co$^ converge uniformly to CD. foregoing observation implies that each invariant state is dose to an equilibrium state of some $ G S. The norm approximation gives a strong control which one can exploit to obtain a stronger Statement of this form. The
Observation 4.
//
coi,
.
.
.
,
o),, G
<^('fi ), then
AO
coi,...,cO;, G
E^^
First note that
is
a
simplex by Corollary
between
isomorphism
E^^
G
CD
and
/i,^ G
MCO ('!') is isometric from the
||co
co'll
||/,i^
I.L^,\\.
there
z^ a
.
4.3.11. We next argue that the
maximal
topology
norm
G S such that
probability measures norm topology, i.e.,
to the
To establish this let
co' co co_ denote the co+ co^ given by Proposition 3.2.7. By uniqueness the must be Z^'-invariant and hence there exist unique maximal probability
Jordan co
=
decomposition
measures
/,ij_ with
uniqueness
/i^ -f-
of
barycenters co/||co||.
||co_||/i_
/.i^^/ +
=
> =
Now let co'
Ik^co
can
/wll
hence co/ G
=
find < ^-
n~^((jü\
B^t this
^(/^^)
-
co
implies
for each
z
=
+
-
llAia;-/Wll co)
h G
co
+
co_
=
co' + GL)+ and hence
by
||a;_||
I|||<^-||A^-
+ co2 H
and
But
||co+||/.i^. Therefore,
||co-co'||^||co^||
one
=
co
A^
that
ll^+ll/^+ll 11^ -II
>
and take
such
that
/i^/({co/})
l, 2,...,
> 0
&
<
n~^
.
By Observation
||a;-co^||<e for each /
=
and
l, 2,...,
3
hence n,
and
77.
Thus although Observation 2 indicates that A(p "often" consists of a single point Observation 4 proves the existence of interactions > for which AO contains many different Z^'-ergodic states. Despite the interest of these general observations they shed little light on the structure of {A^CD; ^ G [R} for $ fixed. The convexity arguments are inherently too weak to obtain details of the phase structure for a fixed interaction äs the inverse temperature ß varies. For this it is necessary to develop more analytic methods. The foregoing results give a reasonably satisfactory description of spatially invariant equilibrium states. In particular Theorem 6.2.42 establishes under quite general conditions that the Gibbs and KMS conditions are equivalent to the principle of maximal mean entropy at fixed mean energy. Equilibrium is a balance between the disorder characteristic of large entropy and the order associated with low energy. The main disadvantage of the theory is, however, the strong invariance assumption. Although many physical Systems exhibit pronounced features of spatial homogeneity the assumption of strict spatial
invariance appears to be a somewhat unrealistic idealization. It would be of interest to understand the entropy principle in a broader setting of 'homo-
geneous' Systems.
There
are
several classes of
physical
models which lack in-
Quantum Spin Systems variance but retain strong spatial uniformity. For impurities or models of spin glasses.
example
305
models of magnetic
A typical impurity model has a translationally invariant interaction > together with an additional one-body interaction ^ which only contributes at a subset of randomly chosen points x,^ G Z^' and which represents the presence of magnetic impurities. The points Xn could be distributed in a homogenous fashion and ^ could be invariant, i.e., ^({x}) 'z^.x(^({0})), but if the Xn are not the points of a regulär sublattice there is no underlying symmetry. One can introduce an evolution T and the corresponding KMS states and also define the mean free energy by exploiting subadditivity, homogeneity and boundedness properties. The problems arise with the states and in particular the mean en=
tropy and energy General
äs
spin glass
functions models
over
can
invariant interactions O and ^. One
from O for set
one
the states.
also be defined in terms of t\vo
translationally
that the interaction energy arises class L^ of subsets X c Z^' and from ^ for the complementary assumes
Lvp, i.e., the energy is given by
//(A)
=
^ ^(X) + ^ ^(X) XeL^,
X&L^j
XC.\
.-VGA
.
For
could be nearest-neighbor interactions and 1$ a ran example, O and chosen subset of the domly nearest-neighbors. If
lattices without
periodicity. Although this appears initially to be a more departure homogeneous theories on Z^' it is not necessarily the case. Quasicrystals seem to have strong regularity properties which are, how ever, not consistent with a strictly periodic structure. Therefore it is appealing to model these phenomena with aperiodic lattices, for example, lattices associated with aperiodic tilings of the underlying configuration space. But the most common lattices of this type do have repetitive symmetry patterns even if they lack complete periodicity. Therefore one can build the theory to take these symmetries into account and consider interactions with the corresponding invariances. Again the difficulty in fully developing the theory is the identification of a suitable set of states whose structure adequately reflects the symmetries on
radical
from
306
Models of
Quantum Statistical Mechanics
and invariances. This
but there is
marks) translationally 6.2.5.
certain extent be achieved
(see Notes and Rereally adequate counterpart for the compact set of
can
no
to
a
invariant states.
of KMS States
Uniqueness
Spin Systems were originally introduced äs models of paramagnetism and ferromagnetism. A material is paramagnetic if it assumes a magnetic moment in the presence of an external magnetic field and ferromagnetic if it is capable of possessing a magnetic moment in the absence of a field. Ferromagnetism is a relatively rare quality and the majority of ferromagnets are alloys, or compounds, of iron, nickel, or cobalt. If a ferromagnet is heated, it loses its spontaneous magnetization at a certain critical temperature 7^, but above TC it remains paramagnetic. Thus, ferromagnets exhibit a type of phase transition; above TC there is a unique state of equilibrium whose magnetization is completely determined by the external field but below TC and in the absence of a field there are various possible states distinguished by the orientation of the spontaneous magnetization. In this and the next subsection we demonstrate that quantum spin Systems exhibit this type of phase structure. This subsection is devoted to the discussion of high temperatures and in the next we examine the more interesting, and more complex, low-temperature phenomena. We only consider interactions for which the dynamics is given by a strongly continuous one-parameter group T^^ of *-automorphisms of the spin algebra ^ and we accept the (T^^, jö)-KMS condition äs a criterion for equihbrium at inverse temperature ß. In Section 6.2.2 we established the existence of (T^^, jS)-KMS states for all jö G [R and our immediate aim is to prove uniqueness for small ß, i.e., for large temperatures. If j5 0, the KMS condition reduces to the trace condition, =
co(AB) for all y4,5 G ^I, and trace
can
co
be deduced
is the
by
an
local
^[x]
commutativity,
o)(BA)
unique explicit calculation
sequent perturbation argument. Let matrix units for
=
trace state
and choose B
and the
on
^. The
which is
uniqueness
a
0, l, e(/;t,/Y), /^^Ä ,A^ be a set of G ^A where x^A. The trace condition, =
.
.
.
identity N
e(i,J,]
-
(A^+
1)-^ Y. ^(i^.k,]e(k,J,] k, -0
give A^
(D(e(i,J,)B]
-
(7V
H-
1)-' ^ 03(e(k,J,)Be(i,,k,]) k,^0 N
=
(N^iy' Y^ di^^j^CD(e(k,,k,]B] k,
=
of the
model for the sub-
=
Q
(N+ir'ö^_^jMß}
307
Quantum Spin Systems Iteration of this argument
yields
(n
\
\i=\
j
<^\^^(i^nh^\-(^^^r^^i.,j., and
(D
If one
is uniquely determined. ^ 7^ 0, the trace property
i=\
is
replaced by
the
(T^,)5)-KMS
condition and
finds
co
e(i.J.]B]
=
(N+\r^ Y. {^(^(k.J.]Be(i,,k,]) k,
+
0
=
CD(e(k,j\}B(rf~i}e(i,,k,}}}\
!/
=
/ iß
^(N-^\r's^..jMB} +
f^ 03(e(k,J,]B(i^-i]e(i,,k,]]
(N+l)-'
k,
=
Q
strictly local elements of ^ are analytic for T^. The second right-hand side of the last equation can then be expressed äs a in series ß, without constant term, and iteration of the resulting equation power allows one to deduce that co is uniquely determined for ß sufficiently small and for a large class of O. The details are äs follows. Each B G '^x can be expressed äs a linear combination of matrix units
Now
assume
the
term on the
/7
e(Ix,Jx}
=
Y[^(^^nJ-^i^ /=!
whereA^ to
=
{xi,
.
.
consider the
Now
adopting
,Xn}Jx
.
=
{/jc,,
special assumptions
the
(T*-z)(efe,^,))
=
{7^, ^J ,Z:.J, andJ;^ e(Ix,Jx]Jx,^x G {0, l, ^
-
choices B
,
of Theorem 6.2.4
;^^^
one
.
Thus it suffices ,
A^}^,
Jf c L.
has
^ C,f({;^.};/,,^0
,
'^
n>\
where
C^({X,}'J,,k,] ^0
=
W,
and
Sj
^
=
[^(Xn]A'''[^(X,]Ai^.k,^^^^
Xj^Xj^i u A^
u
JTi
u{;c}
,
But if
.
\^
^A(Ix^Jx)e(Ix.Jx)
Ix.Jx
is the
decomposition
of ^4 G
^x^
the
complex
\A(Ix.Jx]\
<
coefficients
\\A\\
.
A(Ix^Jx] satisfy
308
Models of
Quantum Statistical Mechanics
Therefore
C^({JG-};4,^.x)- ^ C({Jf,-};/,,^,;/^,J^M/^,J^J
,
fSn.JSn
where
C^({Xi};i,,k,-Is,:,Js,.) l
l
n \M^i)\\
2
<
i=\
Moreover,
e(k,j\,)e(Ix.Jx)Cf({X,}',i,,k,)=
^
a^ (J^. J^. X/^. J^. ) ,
,
/_yX J^X ,
where
iS*;^
=
{jc}
uX
u
S,;, and there
are
(7V
at most
+
l)-!-^"!
nonzero
coefficients
a^^ which satisfy
|*(4;,^5;)l<2"nil
.
/=!
This
perturbation expansion
for co, evaluated with B
{a9(e(/7,/7));
Y c.
L}.
=
This
can now
e(Ix,Jx], equation
be combined with the
previous identity equation for the family interpreted äs an integral equation
to obtain
will be
a
linear
suitable Danach space. Let 3E be the Banach space of bounded
on a
{Ix^Jx}
...
plication together
with
the
complex functions / on the pairs L, and / (/0, J0) G C. The space operations of addition and scalar multi-
,Nf ,X
where Ix^Jx C {0, l, X is equipped with the natural
c
supremum
norm.
If
co
denotes
the
family
{cD(e(Ix,Jx)]]X C L}, where we take e(I0,J0) H, it follows that co G 3E 1. The foregoing identity and perturbation ||co|| expansion yield =
equation co
for
co
where
ö,K,
and
Lß(^
ö(Ix.Jx}
Ä^ is
a
are
=
ö-\- Ko} -\- Lßf^o)
defined
äs
follows: ^
\X\
and
ifX
=
l
(N^ir'^t^j'.
ifX
=
[
0
otherwise
0
{x}
linear operator with action
{xi,...,Xn},X'
=
=
(N+ir'ö,^^^j^J(Ix'.Jx'}
{x2,...,^J,/7
>
2,
and
(Kf)(Ix.Jx)=^ if
G X
r l
(Kf)(Ix^Jx) ifX^
=
<
2,Lß^
is
a
linear operator such that
,
and
the
309
Quantum Spin Systems
(Lß^f](ix.Jx]
E^
(N^\r' V z^^
=
z^ k^^Q n>\
v
v
^\^-^n
4x/5:c
Z-^
I
Z^
A'yn5-_i5^0
xa^(/5.,J5^)/(4^,J^.) a^ arising
where the coefficients ciated with
a
fixed
The above
is
co
K
=
are
asso-
X\{x}.
form
Lß^)a)
ö
=
.
and
uniquely determined, Q}
,
perturbation expansion
^X' with X'
=
(l Hence
from the
Splitting {x} integral equation has the X
"
^(K^Lß^Yö
=
>0
whenever
ll^^oll by
<
use
||^ + Lß^\\ < 1. But \\K\ ll^ll) N(N + 1)"^(l
(N The
=
of the estimates
on
Cf
and the
-i-
1)"^
norm
and
ofLß^
procedure
so
uniqueness
can,
used in
is ensured if
however, be bounded the proof of Theorem
6.2.4.
Proposition
6.2.45.
i|a.||,
Let O be
an
interaction such that
^e^"(A^+i)'"fsup
=
for
some
A
>
Y. mx]\\\< +
V^^^--,
^0
^
/
0.
It follows that there exists
a
(T:^,ß)-KMS state
unique
whenever
i/>ii*ib<ß)('.+-''<''""'^" 2/1
PROOF.
The estimation
definition of
procedure
N
used to prove Theorem 6.2.4
together
with the
Lß<^ gives
\\Lß<,\\\
^ ni,...,n
[]][(,
+... + ,._,
+
i)lia)i|_ (7V + 1)2"' VA^ + I)^
\/=]
,
/
where
^
||
11$ W II
.
A-9.t
m=+i
But /;
[J(i
n
+ /i2 +
+ /-l
+
1)
<
(l
+ 2
+
+ /!
+
1)"
<
r"n \e'-
JJ e^-^^'
.
310
Models of
Quantum Statistical Mechanics
Therefore
.^||<(A^
whenever
(2 |j5|//l)||||;^
the theorem after
<
l)Vffl)||0||Yl-[^)||Ol|,
+
1. The stated result foUows from the discussion
preceding
simple rearrangement.
a
Proposition 6.2.45 establishes uniqueness of the equilibrium state whenever \ß\ ll^ll;^ is sufficiently small. We have made no attempt to optimize the estimate on |^| and this can certainly be improved in numerous ways. First, if
^(X)
=
0 for
\X\
>
m more
efficient
given interaction the bounds
norms can
be introduced.
Second, for
any
the
important lowest-order terms in the perturbation series for L^$ can be individually calculated and this may significantly improve the total bound. Third, it is possible to exploit other matrix parametrizations, i.e., if A/' l, each 2x2 matrix is a linear combination of Pauli matrices and this parametrization can lead to better bounds. Thus Pro position 6.2.45 should only be taken äs a general qualitative Statement. For physical applications to magnetic Systems this proposition is delinquent in one important respect. It does not establish uniqueness for small ß and all external field strengths. To obtain this result it is necessary to use two additional techniques. First, one handles the one-body interaction corresponding to on
=
the external field in
lowest
a
separate
manner
and, second,
eigenstate reexamine the integral equation for co. (I)(^) +O^ where $^^^ is Let (I) =
one
must
of this field. To introduce the first of these
^({x}),^^^\X)
0 if
1^1 ^
the
one-body
separate
techniques of
part
out the
we
must
,
i.e.,
l, and ^' is the
multiparticle inter action. We again assume that determines a one-parameter dynamical group T^^ for which the strictly local elements of ^ are analytic. The one-particle interaction )^^^ automatically determines a second group T^^ with the prop=
=
^l^.^^ for all x erty that T^^'^(^|_^|) T^^^^ ^commute. Therefore, t e U^^it
e
=
=
A G
L, and
^fr^^^/^
is
we a
further
assume
T* and
one-parameter group and for
^IA,A cL,
di,(A) =
E
/T/
[^'W'^]
\A'nA^0 It follows
readily
that ^' generates
Next consider the
a
/
dynamical
group
T^'
and
-
Now suppose that the matrix units and let c/),(z),/ 0, 1,2,
(I>(^)({jc}) arranged
correspond
^
.
in
i^'
^f^^-t
identity
o,(e(i,,J,)B)=oi(e(k,j-,)Bifß\e(i,,k,))) + co(e(k,,j,,)B{,fß T!;.; ^)Tf;" (e(/
has
that
increasing
order
.
.
to an
^,)))
.
eigenfunction
,A^ denote the eigenvalues of
repeated according
to
multiplicity.
basis of
^^^\{x}} Then
one
311
Quantum Spin Systems
T^^'V^r/ k}}- ..-^(^.(^-)-^x(^.)),,f/ k } (^Vx^f^x))^ ^Vx^l^x) ^iß
'
Therefore
^/^(^.(0-^.(^.))^(^(/^^yj5)
=
di^j^cD(e(k,, k,]B]
-^CD(e(k,J,]B(^fß~i)e(i,,k,]] for all B G
^A with
jc
A.
^
Summing
k^ yields the identity
over
co(.fej-.)5)^ff^.^(^^(^-^)-^^(^-^)^^/ \^.=o
f
X
^
<^l d,^j^CD(B) ^ co(^(^A)5(Tf^' -h
-
Ä:,=0
This
of identities
family
be translated into
again
can
the Banach space 3E. The major dependence into the extra factor 02
on
new
l i)e(i,,k,]] \ J
.
integral equation for Separation of the O^^^-
an
feature is the
> ^-'
7V
ß((p,(/,)-(^,(^,))
,E^' Ux
=
o
A moment of reflection establishes that if this factor is
the
integral equation But if jß > 0,
small in
are
f l
^'^
0 and
=
=
now
The
^x
^
new
^{x}
can
solution.
(^^^(ö) < ^jcC^)- Similarly if j5 < 0, the factor is less than 1/2 (px(N) > (p^(N 1). In both the exceptional cases, how-
take values
concentrate
the
"large"
arbitrarily
idea is to
possible
the
on
dose to
one.
case
jS
0 and describe
>
0. The
parametrize each
e(Ix,Jx]
A e
in which
case
=
find
a
terms
l-f^e(M-)
.
^'A(Ix,Jx}e(Ix^Jx) Ix,Jx
a
,
method of cir-
0 is handled simi-
with i^
decomposition A=
<
^x in
/= l
one can
^
e(ixjx)
because
e(0,0)
Thus the factor remains
two situations.
behavior for /jc
ai^d matrix units
cluded. This is
Thus
unique
2 ^
l
small, i.e., less than 1/2, in all but
cumventing larly.
a
N and
ever, the factor
We
has
i1
<
/
o
"small," the kernels of
equation
\~^
N
\^.
except if ix
and the
V e^('^-^(^'-)-^;c(^.)) l
l
except if /jc
norm
of the
identity
0 =7'^ is
ex-
312
Models of
where the
prime
Quantum Statistical Mechanics
denotes the exclusion of the (^
=
0
=
j^ units, and it is readily
checked that
\A(Ix.Jx)\<2\^^\\A\\ Note also that the number of terms in this
.
decomposition
is
(7V +
1)"'^'.
Now introduce the Banach space 3E' of bounded complex functions on the 0 pairs of indices {Ix.^x} with the i^ jx,x G X, possibilities excluded. The =
State
CD
defines
an
element
{o}(e(Ix,Jx)}]X
CD
L}
c
of this space and the state
is determined
1. Moreover, if by co and the normalization condition co(1]) one takes the perturbation expansion for T*' ^ in the new identity for co, the resulting relation is interpretable äs an integral equation on 3E^ This equation =
-
has the form Q}
=
^-\-
Kß^(i)0} + Lß^co
l
ifX
=
0
\-'
^ eßMi,)-v,(i^,)} \ k,
=
{fX^{x}
S.^^j^
J
0
0
and
Kß^(\}
is
a
otherwise
/
(Vo/)(/.,Jx)
^
,/^K(^. )-/>.. (^.))
E
=
=
{xi,...,x,,},X'
=
Y' ö,^j_^j(ix'.Jx'}
J
V^.=^ ifX
,
linear operator with action
{x2,...,Xn},n
> 2
and
(Vn/)(/x,^^)-0 if
\X\<2. Therefore,
||^^^(.)||
-
if
ß
>
0,
exp
{ß((p.,(l)
supf^^(^/'-^(^)-^-^())
-
9:^(0))}
E
+ ^ +
xeL
k,
=
>
l and
e-ß^^P-^^^-^^-^^-^-^^A
2
l -2
Thus
||^^(D(i) +^/:?CD|| <^+ ll^/itDÜ and there is
a
unique
solution to the
integral equation
\\Lß<,\\<\
whenever
.
Now the operator Lß(i> is similar to the operator occurring in the previous equations and its norm can be estimated by a slight modification of the earlier
Quantum Spin Systems estimates. The
new Lß^ differs from the old replaced by O'. Second, an additional factor
one
313
in four respects. First ^ is
(7V+l)j^^^/^(^.(/.)-
==
terms. In
jx
~
/
0
Third, the action of Lß^ is slightly modified by the omission of the ix particular the perturbation expansion of
occurs.
^
=
efcJ^)(4-z)(efe,Ä,)) contains terms
proportional to e(Ix^Jx)e(Is,Js^) and e(0,0) terms may occur through products e(0, 7)^(7, 0) if ^ n S 7^ 0. To eliminate the e(0, 0) we must use the identity N
e(0,0)-l-
^e(^,^) k= l
and hence in the
expand into (A'' + l)'*^"' terms. Fourth, the coefficients possibly larger by a factor 2''^"' and these last two effects combined factor (2(N -\- 1))^^"^ in the term-by-term perturbation
one
term may
expansion
introduce estimate
a
Hl/j^^ll-
are
Thus
one
obtains
IM<^-2.( where
we
+
i)V(f)im(,-(f)|i*'
have assumed that
\m,
=
^ 2"e^''(N + 1)^" sup ^
\\^(X}\\
<
^^
'^\f-., for
some
/l
>
0.
Therefore,
A
if / T\
ß\m,<(jj(i+2(N-^i}V
\4^/l\-l
has
\\Lß^\\
^
and hence 02 is unique. may be repeated with the i^ The final result is summarized by the following: one
If
<
ß <0, this argument
Theorem 6.2.46.
(D(i)({;c})
-
a)({jc})
Let O be
and
an
^^^\X]
interaction with -
0
=
N
=
jx
terms omitted.
one-body part O^^\ i.e., Q)' O^^^ and ^
if\X\i^\. Define
=
-
assume
||$'||,=
^2V(7V+l)^sup ^
for
some
/l > 0. Further
mx)\\<
''\r=f;.,
^'
assume
that x^
and T*^ commute.
+cx>
Models of
314
Itfollows
Quantum Statistical Mechanics
that there exists
a
unique
^
\
/
(i^ ,ß)-KMS ^x4
x^/,^
^
ß\m,<(-^j(l+2(N+l)'^e Note that
no
uniform bound
for all
||
on
and all
l
is needed.
strengths inhomogeneities The integral equation method establishes more Both integral equations have the form
occurs
A\
whenever
state
Thus, uniqueness
of the external field.
(ß
=
Ö-^
than
have
we
actually
stated.
Mß^CD
and for the interactions described
the
unique (T"^', ^)-KMS
state
by Proposition 6.2.45 and Theorem 6.2.46 co^"^ is determined by
^^'^(e(IxM}
CD'
=
((^-Mß^^'mx.Jx) \~^
-
'^(M'^,,S)(Ix,J;,)
n
>
.
0
But if
ß^ varies within the appropriate domain of uniqueness, one may ß<^ \-^ CDf^'^(e(Ix,Jx)) is continuous, and hence ß^^co^^(A) is continuous for each ^4 G ^A and A c I. A similar argument for continuity of T^^ was outlined after Example 6.2.5. In fact, this estimation procedure estabhshes that if ^(O + jci^iH +;c,jO) is in the region of uniqueness for some >, Oi, )2, , and all (x\, JC2, ,;c) in an open subset of R", deduce that
.
.
.
then
(jci,...,x,)^co^^'^+''^^'+-+^^^)(^) extends to Next
a
one
plications
to
holomorphic
function of
variables, for each strictly local A. paramagnetism. In apthe interaction is usually of the form magnetic Systems one-body
may deduce
some
n
basic characteristics of
0(^)({4) where h^ G IR is the external field
=
-/z,I'5 and
^ is the third component angular corresponds to the couof the atomic and moment an pling magnetic electromagnetic field in the direction of the third coordinate axis. With this interpretation the mean magnetization of the atom at the point x- is given by of the atomic
strength
at
x
momentum. This interaction
M(ß,h,)
=
CDß'\Ll)
and the considerations of the
previous paragraph, coupled with Theorem M(ß, h^} extends to an entire analytic function. Now 1.3 has 7V + l distinct eigenvalues / A^, in suitable units. N/2, / 0, l If e(ixj'x) are matrix units defined by a corresponding eigenfunction basis, the identity for co^^ used in the proof of Theorem 6.2.46 implies 6.2.46, estabHsh that h^
e U^-^
=
,
.
co^'^(e(4,4))i
.
.
,
,
315
Quantum Spin Systems where
Cß $'
Thus if
j8
>
positive constant independent of the external the temperature is positive, i.e., 0, is
a
lim
co^^(^fe,O)-0
ifi^^N
cü''*(e(/;/;,))=0
if/^T^O
field
strength.
hx^+oo
and
lim hx^
.
oo
+CXD the atom at site x is in the eigenstate of 3 Consequently, äs h^ oo the eigenstate eigen value N/2 with probability one and äs hx eigenvalue -N/2 plays the predominant role. Explicitly one has
with
>
with
-^
lim
M(ß,hx)
=
hx-^ 00
and this demonstrates that for
large
^
2
external field the atoms
align
themselves
with the field, i.e., the System is paramagnetic. Therefore, one concludes that in the high-temperature regime spin Systems are paramagnetic but not ferro-
magnetic. There
are
several notable corollaries of uniqueness of the
co^*. First, 60^^ is automatically an extremal factor state by Theorem 5.3.30. But if Z
(T^, j5)-KMS
state
(T^, jS)-KMS
state, and hence a Z^, the spin algebra is norm the group Z^ of space translations. Thus ==
asymptotically abelian with respect to o)^* is strongly clustering (strongly mixing) in space by Example 4.3.24. Next if O is Z^-invariant uniqueness implies that co^* is Z^-invariant and hence strongly clustering (strongly mixing) to all Orders by another application of Example 4.3.24. Next assume that the finite-volume dynamical groups T^'^ converge strongly to T*. Uniqueness of co^* then implies that the finite-volume Gibbs States co^^ must converge in the weak* topology to co^^ äs A oo. This is of follows because each weak*-limit point 0;^^ a (T*, jß)-KMS state by Proposition 6.2.15. Therefore, all weak*-limit points must be equal to CD^^ and -^
hence the limit must exist. There is another
general
uniqueness of the equiuniformly bounded surface uniqueness is independent of the
class of models for which
librium state is usual, one-dimensional models with
For these Systems, however, the temperature and hence no form of phase transitions
energies.
dimensional
models
with
bounded
surface
energy
can
do
occur.
not
Thus
exhibit
one-
ferro-
magnetism. Theorem 6.2.47.
Let O be
an
interaction
satisfying
Y,mx]\\<+^ X3X
and
assume
there exists
an
increasing
E X
r^^ni^
Xr^k^ ^ 0
sequence
^(^^
<
M
A
c L
such that
A
^
cxo
.
316
Models
for
of
Quantum Statistical Mechanics
Further, let T^ denote the corresponding aiitomorphism
M < +00.
some
(constructed in Theorem 6.2.6). Itfollovvs that there is a unique (i^ ß)-KMS
groiip
,
PROOF. with
state
all
for
ß
^ U.
As this result is not central to any subsequent results we content ourselves proof which uses the ^* -Version of the theory of perturbed
outline of the
an
States described in Section 5.4.1.
(=co^J
First, let WA,,,
be
weak*-convergent subsequence of the local Gibbs an arbitrary extremal (T^^, ß)-KMS state. The idea is to prove that co co^ and hence deduce that co is unique. To accomphsh this we introduce the perturbed states CD^^"' defined in Theorem 5.4.4 but rescaled to ß, with the perturbations P/ given by States with limit co^
Next, let
a
be
co
=
,
Pn'
W^(M
=
Y^
=
^(^)
.rnA,^0 A'-', ^ 0
X
n
It follows from Theorems 6.2.6 and 6.2.18 that
hence the restriction of co^^"' to of
co^^"'
'
äs
happens
^
because
and hence
äs
A/
^A^,
is
co
satisfies the Gibbs condition and
co\^, Therefore, co' is the weak* -limit Next, we must prove that co^^"' converges to co. Basically this and co^^"' differ only by a perturbation across the boundary of A/
equal
to
00.
co -^
oo
this difference becomes
negligible. Unfortunately
the
proof of
convergence is rather intricate.
The
key
to the
proof is the introduction of the family of functions
p (^. A\ ^n(-^t^)
-
(p-i
:
where
(^,71^^, L/co,^^^) denotes the representation associated with the i^-invariant state CO. It follows from Theorem 5.4.4, and its proof, that each F/^ is holomorphic in the
Strip T)^/2; Im z G (0, ß/2} and is strongly continuous and bounded on the closure of the Strip. The functions F interpolate between co and its perturbations co^^"' For example, one has
X)^/2
.
Fn(t ',A)
(F:f"n,,7i(^)Fr^"Q,,)
=
,
and
F(t when
^ G
+
iß/2 ,A)
=
(a^^?(^"), rr^-;.4^)r-^"nr '''"')
R and
p-P. is the
unitary co-cycle
^
^it(H-n,,(P)}^-itH
associated with the perturbation of
F(0 ,A)
=
0}(A}
Utjj(t}
=
exp
{itH}.
Thus
.
Moreover,
Fn(iß/2 -A) Next
=
co''\AmßJ"f
=
o,ß^^(A]Fn(iß/2'^]
.
one argues that the functions t e U^ Fn(t^ iß/2 ]A) and t e U^ Fn(t ]A) uniformly bounded and equi-continuous, i.e., continuous uniformly in n. The uniform boundedness hypothesis of the theorem, ||P|| < M, is essential for both these properties. Consequently, one may choose a subsequence n'' of the sequence n' are
such that the limits
Quantum Spin Systems lim
F{z-A)=
^f
n
,
oo
lim
F(z-^)=
F(z;A}
317
F.(z;1])
,
n"^oo
exist for all
T)^/2 and ^ e ^. In fact, the convergence is uniform for 1)ß/2 and z\-^F(z]A) and zi-^F(z;1l) are holomorphic strongly continuous and bounded on T)^/2e
z
subsets of
Next
reinterprets this convergence in 0 for allAe^ and hence
one
II K '^111
-^
lim n'^
But the convergence of F//
weakly an
and the limit is
Thus
(r:f"")T~^''"
in compact
in
T)^/2
and
operator form. One argues that
\\[r-^"',n^(A)]\\=Q
.
00
to F
R then
on
automatically
(T^,^)-KMS
extremal
z
an
state this center
converges
weakly
implies
that
(rif''")*r7^""
element of the center of
to
7Cf^(5l)''.
converges
Since
co
is
is, however, trivial by Theorem 5.3.30.
F(r ;1)1](-: 1).
F(t',A)
=
o}(A)F(t',1\)
F(iß/2',A)
=
co(A)F(iß/2',^)
Hence
and
by analytic continuation. one
But from the identification of F
and the Gibbs condition
has
F(iß/2-A)= limFn"(iß/2-A) n"~' =
lim
o/'''"(A)F..(iß/2;^)
=
(a'(A)F{iß/2 ^)
.
fj/f-^OQ
Thus
(co(A)-CD\A))F(iß/2',^) and to conclude that the
co
=
=
CD' li suffices to prove that
Q
,
F(iß/2
;
11
) 7^
0. But if
we
apply
Peiels-Bogoliubov inequality
F(iß/2;^) established
=
||af"||^
prior to Example 6.2.30, hypothesis.
>
exp{-ßco(P)}
the desired conclusion follows from the uniform
boundedness
6.2.6.
Nonuniqueness
Our next
topic,
the
of KMS States
nonuniqueness
of KMS states at
sufficiently
low tem-
perature, is of paramount importance for the explanation of phase transitions in magnetic materials. Physically, the nonuniqueness implies the existence of several distinct a
thermodynamic phases. But at high temperature there is usually unique phase, by the results of the previous section, and hence the phase
structure varies with
There
temperature and transitions
least two
can occur.
phenomena which lead to phase transitions in paramagnetic materials, ferromagnetism and anti-ferromagnetism. The first of these has already been mentioned. It is the ability of a material to retain a are
at
318
Models of Quantum Statistical Mechanics
residual
magnetism
in the absence of
an
external field. The second is
physically
less apparent and is detectable from the behavior of the magnetic susceptibility This is given by ;( M /H, where M is the overall magnetization and H is the ,
=
external field
strength. The susceptibility tends to increase äs the temperature ferromagnetism is signaled by a divergence of ^ for small // at a temperature T^. In contrast the susceptibility of an anti-ferromagnet
decreases and critical
increases to is
a
finite maximum at
lowered, i.e.,
i has
a
cusp at
TC and then decreases äs the temperature Thus, below the critical temperature, anti-
some
TC
.
ferromagnet s tend to resist external magnetic forces. The simplest phenomenological explanation of the spontaneous magneti zation of ferromagnets is given by postulating that the interaction favors the ahgnment of neighboring atoms. The alignment of the magnetic moments of a sufficiently large proportion of the individual atoms gives rise to an overall magnetic moment. Similarly anti-ferromagnetism can be explained by supposing that neighboring atoms prefer to be anti-parallel. Unfortunately, these explanations are insufficient. For example, we have already deduced that onedimensional Systems with uniformly bounded surface energy cannot have multiple phases. Therefore the cooperative effects of alignment or misalignment of the magnetic moments is not explainable by purely energetic considerations. One way of
understanding
this
difficulty
is
through
the
principle
of
max
imum entropy. This principle characterizes equihbrium by maximum entropy S at fixed energy H^, i.e., by maximization of the function S ßH^. Thus, for
positive temperature, ^
>
0,
two
competing effects
occur, maximization of the
minimization of the energy. Large entropy is associated with entropy disorder but low energy is favored by the orderly alignment, or mislarge of alignment, neighboring atoms with the particular choice dictated by the sign versus
of the interatomic forces.
ferromagnetism, interplay of these
Thus, the existence of ferromagnetism and antiphase structure, depend upon a delicate
and the details of the two effects.
Detailed theoretical
understanding
of
ferromagnetism
and
anti-ferromag
netism is based upon the study of specific models within the class of general spin Systems and it is now necessary to specialize considerably. The simplest
quantum System is the We examine the
ficiently
spin-^ Heisenberg
anisotropic
model introduced in
Example
6.2.2.
Version of this model and establish that for suf
low temperatures and suitable values of the
anisotropy there are at analysis is to view classical limit, the Ising model.
least two distinct extremal KMS states. The idea behind this the
Heisenberg model äs a 'perturbation' of its anisotropy plays the role of perturbation parameter. We emphasize that a great deal of work has been done on particular models and we make no attempt to survey the subject. We merely try to describe the two most common phenomena which lead to multiple phases and some methods used to elucidate them. The classical Ising model provides the basic insight into these phenomena and plays a key role in understanding the more complex quantum mechanical models. The
319
Quantum Spin Systems Let L
=
I.^
.
The
spin-^
quantum algebra 91 {j^} consists of linear combina-
tions of the Pauli matrices
classical
corresponding (73. More generally
the
in the Pauli matrices in
defined in
ÖQ(= H), cr^, o^-, 0^ algebra G{^} consists
spin-^
{öf ; z
Example
4.2.7 and the
of linear combinations of OQ and quantum algebra 2lA is generated by polynomials 0, l, 2, 3,;c G A} and the classical algebra CA by
==
polynomials {öf;/ 0,3, nearest neighbor interaction =
jc
G
and
A}.
anisotropic Heisenberg model with given by the interaction
The
zero
external field is
O where =0
if
|;^| ^
^({x,y})=(^
if
\x-y\^\
^(X]
^({^. y}}
-
-X^, y}{^(^y.
H-
^2^2^
+
^A]
if where
|jc|
troduced
==
by
|jci|
4-
and
\x^\
-i-
the addition of
a
j(x^y)^d
G [R.
\x-y\
=
\
An external field
one-body interaction,
^(^\{x})^-h(x)a\
can
be in-
e.g.,
.
The Parameter d measures the anisotropy. The value ^ 0 isotropy and ^ corresponds to the classical =
=
l corresponds to Ising interaction
j(x,y]o^^a^^ G ^{x.y}- Note that 7(jc, j;)ö^cr^ viewed äs an ele^{^^,3;} ^4 has two doubly degenerate eigenvalues 7 (A:, y]. The value
({x,y}) ment of
2
=
occurs if a\ and 0-3 have the same eigenvalue, i.e., the atoms at x and y aligned. Thus the model is defined to be ferromagnetic if 7(;c, j^) > 0 since the aligned state is then the state of lower energy. Similarly the model is defined to be anti-ferromagnetic if j(x^ J^) < 0 since the non-aligned state is then energetically favored. We assume that the real functions j and h are uniformly bounded and then the Heisenberg interaction O defines a dynamical group of *-automorphisms T* of the spin algebra ^ by the theory of Section 6.2.1, for example by Theorem 6.2.4. Our aim is to establish the nonuniqueness of (T^,ß)-KMS states for sufficiently large ß and small anisotropy. First we discuss criteria for non uniqueness.
-j(x^ y] are
The first criterion is the existence of If /z
=
0 and there exists
then there is state is
more
given by
a
than
a non-zero
spontaneous magnetization.
(T^, jS)-KMS state co with co(ö^) ^ 0 for some x one (T^, jS)-KMS state. For example, a second
a*a; where a* is the
adjoint
of the
*-automorphism
a
G
Z''
such
of 21 with
the action
aK) for all
jc
=
cT^,
((7^2)
=
G Z^. This follows because ai^
-'^i, =
(^)
=
i^a. We refer
-^ to
a äs
the
spin
re-
fiection symmetry. Note that if there is a (T^, jS)-KMS state co with O3(a\] ^ 0, or co(ö2) 7^ 0, then a similar argument establishes that there is more than one
(T^, jS)-KMS
state
even
if /z
The second criterion for This
gives
an
inconsistency
^^ 0. nonuniqueness
is the existence of
between a-invariance and
long ränge order. extremality. Assume that
320
co X
Models of
is
(T*^, ^)-KMS
a-invariant extremal
an
Ti(j3(^)"
Z\ But
G
Quantum Statistical Mechanics
is
factor
a
by
state. In
particular cü(cr^)
Theorem 5.3.30. Moreover 51 is
=
0 for all
asymptoti-
cally abelian with respect to the group x G Z^I-^T^ of space translations. co is strongly clustering in space by Example 4.3.24. In particular lim (ß((r}(j^^) |j;H 00
Thus, if for
a
lim l vH 00
=
{a)((r](7^)
given a-invariant (T^, jß)-KMS iiif x,yel.''
then
state
^(^\o\\ ^
>
one can
model for suitable values of
.
deduce
0
state. Hence there must exist at
least two distinct extremal KMS states. We will
Heisenberg
0
^
(T^,j5)-KMS
cannot be an extremal
co
(j^((f^<j^((j\}\
Hence
ß
and ö. It
verify this criterion in the corresponds to long ränge
Order in the System. Theorem 6.2.48.
Let ^ denote the ferromagnetic
nearest-neighbor coupling j(x,y) ternal field. Assiime v > 2 and j
inf
=
Heisenberg
0, anisotropy d
>
/(^, j)
>
0
G
interaction with
(1,1)
and
zero
ex-
.
(x. y]
It follows that there is
(T^, ß}-KMS
States
a
all
for
ß^ > 0 ß > ß^.
such that there exist
at
least two extremal
Note that this result does not establish
nonuniqueness of the KMS states for 1. In fact if v 2 the criterion of nonisotropic model, i.e., for the case ö zero spontaneous magnetization, fails 0, by symmetry arguments. co((r^) ^ the
=
=
This will be discussed in detail later in the section. We will not
(5
give
a
complete proof
of Theorem 6.2.48 except for the
special
0, the Ising model. We explain, however, how the ö 0 result can be extended to small (5 by a routine perturbation argument which does not depend case
=
=
upon the detailed form
give
a
((r|cr[
brief discussion of the
dose to
one.
The
Observation 1.
+
cr^cr^)
more
starting point
is
of the transverse interaction. Finally we sophisticated argument used for values of ö the following observation.
Let 0 be any interaction such that
reflection symmetry
a.
If there
exists
sup
an a
invariant
CO(P-IP^_^)
<
T^
commutes
with the
(t*^, ß)-KMS state
1/4
co
spin
\vith
,
x,y e Z'
where
P^^
=
(H
0^3)72,
then there exist
least two distinct extremal
at
(T^, ß)-
KMS states.
This is
just
a
rephrasing of the criterion
of
long
before the Statement of the theorem since
co(P-;P^) by
reflection symmetry.
=
1/4
-
(o(a\a^,)
ränge order established
Quantum Spin Systems Next let a)\ denote
for all
G
z
Z^'
one
a
state on the local
CD^
=
(P\py_
n z
expand
the
the
product on calculating an
which is useful for
terminology
P-_^
-f P!_
=
H
may write
o^^(P\P^L] We will
Because
algebra ^A.
321
e
(P;
+
Pl))
.
A\{xj;}
right-hand
side and obtain
an
identity
upper bound. We must first introduce
some
and notation.
NN(A) denote the set of nearest neighbors in A, i.e., the set of ordered 1. If ^ C NN(A) is a subset of pairs (x, y] with x, j; A such that |;c ^1 C that is ^ A U~ set connected if for any two points x,y ^X we a NN(A), say there exists a chain {jci \ y] m such that (:Cj,x/+i) G U and jc, ;c2, ,Jc+i for (7 A z contour y is a subset of NN(A) with the foll, 2, (xi+\,Xi] G lowing properties: Let
=
=
=
.
.
.
=
A is not
(1) (2)
NN(A)\'y-connected.
There exists two nonempty subsets Ai(7) and A2(7) such that AI (7) and are AI (7) NN( A) \7-connected, Ai(7) n A2(7) 0, and =
AI (7) If
(3)
u
(jc, y]
A2(7) G 7,
=-
A.
then
%
G
AI (7) and
A2(7).
y G
Hence, if A is NN(A)-connected, there is tween ordered
A2, and
partitions (Ai, A2) NN(A) given by
of A into
a
correspondence be-
one-to-one
NN(A)-connected
subsets AI and
contours 7 C
7-{(x,;;)GNN(A);zGAi,3;GA2}
.
We will
always assume A to be NN(A)-connected. configuration c is a function on A with values uration c and two points jc, >^ G A such that c(x) A
in
{-h,}. Fix a configLet r(c) +,c(y) with {(^I,:FI ),-.., (:c,j)} c(jc/) +,c(j/) largest NN(A) -connected subset of A such
denote the class of all contours 7 ^i l, 2, ...,. Let AC be the
=
-
=
that
7(c)
jc
G
AC and c(z)
=^
-}- for all
z
G
A^. Then there exists
a
unique
contour
{(^i,ji),...,(^,;^)} r(c) such that zi,...,x G A^ and x G Ai(7(c)) and y G A2(7(c)). Even though A^ C Ai(7(c)) it can happen that the two sets are unequal. This is illustrated by the second of the following two figures, which a
G
=
give examples of how j(c)
is defined for two different
configurations
c on
two-dimensional lattice. +
+ -++
+
+
_i_ l +
4+
-4+
+
-
|- ph
+ +
In these
examples, the
the indicated
ponent
curve
occurs
first.
contour
y(c)
in H
+
-fe + +
is the set of nearest
and ordered such that the
point
-
-
-
+
+
+
neighbors separated by
in the x-connected
com-
Models of Quantum Statistical Mechanics
322
Using these definitions
n^^-
we now
have
Enn(-^)
=
c
=
E
E
y
{c;y(c)=7}
ze A
n^-^(.)'
where the
P^
sum is over all configurations c with c(x) mutually commuting projections and hence
are
=
n ^^w
0 ^
for all
whenever 7
c
E {c;y(c)
A.
C ^ C
y}
zGA
E {^ ; r(c)
deduce the
Observation 2.
following För any
c(y)
.
But the
1
n^^.(.) 9
zGA
y}
]][ (5, 0
we
+ and
ZG r
-P';p^ and
<
A
Therefore,
n^c(.)^
=
n ^c(.)
^
zG^
zG
p^^pL
n
<
(s, 0
G y
^+^-
G y
bound.
region A
P"+P--
/z^
ö^e
Y^Py
<
y
where
py=
U (s, t]
and the
Next
sum
is
over
n^i
G y
all contours y with
x
G
AI (7) and
y
G
A2(7).
counting argument, commonly called the Peierls to most of the current proofs of phase transitions in spin Systems. It gives a quantitative assessment of entropy loss versus energy gain introduced by an ordering of the System. Let |y| denote the number of nearest-neighbor pairs in y. In order to bound we
use
a
contour
argument, which is fundamental
the above
value of
sum
|y|,
in the state COA
at least for
large
we
must estimate the number of y with
A. For this
a
fixed
geometric argument in which partially illustrated by the pre-
we use a
implicitly assume v > 2. This argument is ceding figures. To each nearest-neighbor pair (s, t) G y we associate the (v- l)-dimensional hypercube centered at the midpoint of (s, t) and oriented perpendicular to ^ ^ These hypercubes form a hypersurface which separates x and y in A. Now one can construct this surface by successive addition of hypercubes. If one hypercube is fixed and a face specified for the positioning of an adjacent hypercube, there are at most three possible orientations. Given a hypercube on the hypersurface and a face on it there are two possibilities, either it is possible to attach another hypercube to it lying between a nearest-neighbor pair in A, or not, according to whether the face hes in the interior of A or on the boundary. In the first case it follows from the definition of a contour that another hy percube in the hyperplane is actually attached to the face, but not in the second
we
-
323
Quantum Spin Systems
case.
Thus, by
ways of
exhaustion of
an
building
possibilities argument,
up the remainder of the surface
there
once
at most
are
the first
3''^'"^
hypercube
is
given. Now
assume
that the linear dimension of A is
neighbors parating x and if
l^j
large compared
to the dis-
x and y, and let {(x^x\), (jci,X2), j)} be a set of nearest , (jc, minimal of and cardinality. If 7 is a contour seconnecting x y
tance between
.
.
.
and y, it follows that one of the pairs (x/, jc^+i) is contained in 7, given we must either have i < \y\/2 or n i < \y\/2, i.e., the
is
to y must be built up from one of the hypercubes between x/ and x/+i, where i satisfies the stated conditions. It follows that there
hypersurface corresponding are
at most
M3M-' admissible contours of obtains the
Observation 3.
contour is 2 v
one
is any state such that
If COA
COA(P,) for large
length of a
As the minimal
length 1 7 1.
following.
A and for
some
K >
log 3,
coA(n^^)<
e-^W
<
then
^^/-i.-/^ ^n^-^e/"-2v
Up
to this
assume
point
that COA is
the state COA
a
^A- Then if A contains
y and
COA(Py)
=
<
by
the
(T^, jS)-KMS
mutes with
A
^
00.
a
and
Then
co
co
is
on
(T^, jS)-KMS Uy
MA has been arbitrary. Now, however, we T^ of *-automorphisms of
state for a group
is any
unitary
element of 91 A
has
CD^(PyUyU;Py)
CO^(U;Pyr^ß(PyUy))
<
condition. Next suppose the is an a-invariant weak* limit
a
one
(T, jS)-KMS
\\T^ß(PyUy)\\ of the T^
norm
limit
point
of the COA, in the limit
T
com-
state and
Co(P,)<SUp||Tf^(P,f/,)||
.
A
Hence if
one can
establish suitable bounds,
jiiniir^i'p TrM\ ^^P Ir/jßl^y^yJll for the local evolutions
uniqueness
of the
Uy appropriate Let
T^ arising from
(T^, jS)-KMS
and 3. This outlines the
< f^-ß^ß\y\ ^
5
given interaction then nonby combination of Observations l the problem is to make a choice of a
state follows
general
tactic and
for the interaction.
T^'^ denote the local evolution associated with the Heisenberg inter
action and choose
Uy=
n '^i-u; xeAi(y}
.
324
Models of
Quantum Statistical Mechanics
This choice is such that the
Uy implement
the
spin reflection symmetry
a
for
^A,(y), i-e-, a(^) for all ^
G
UyAU;
=
^A^(y).Then
Tf^'^(p^t/,) -^-^^^>(^)p,^^^''^^(^)^^^, But
UyH^(A)Uy
differs from //CD (A)
neighbor pairs (s, t)
G
y.
only Specifically
UyH^(A)Uy
.
in the interaction between the nearest
//0(A) ^Hy-Cy
=
where
^
.//,^2
(^, t]
a),(.,0,
C,
=
^
2
(5,0
7
7(^,0
e r
and
(D5(^,0=y(^,0{('ö+^3^3) ^^24} +
Therefore
Tf^'^(P^^) =e-/^^<^(A)p^/(^.(A)+^v)^^^-^C, Thus
one
_
has bounds
ll^^'l^y^y)!! <^"^^1k"^''^^^^^y^^'''^^^^^^''ll This last A. It is
One
expression is the key to obtaining bounds on ||T^'^(PyC/y)|| uniform in particularly easy to handle in the case ^ 0, i.e., for the Ising model. has decompositions =
//^(A)
=
FA
+
dT^
Hy^Vy^ dTy
,
where FA and
Vy are the Ising contributions to the energies and TA and Ty arise from the transverse X-Y part of the interaction. Therefore FA and F, are functions of the commuting variables a^. But Py is the projection onto the eigenspace
with cr^cr^
=
l for all
r^'Af(^yUy)\^^Q P IJ\\ I.ß and
one
-
(s^t)
G
y. Hence
VyPy
-ßCy ß-^'^AP JV^-^ßVyTJ r^e UyC e
-
=
0 and
^-ßCy
p U ryUyC
has bounds
ll4'(^yt/r)IUo
<
e-P^^^
<
e'^^yl
.
Combination of these estimates with Observation 3 establishes that for 0 the a-invariant Gibbs states COA satisfy the bounds sup
co^(Plpy_}
<
tinct
ß,ß>ßc-
/ > 2
in A. Then Observation l
extremal
(T, j9)-KMS
0
3^/exp{-/(2jß7-log3)}
x.yeZ'^
uniformly
=
states
implies
for
the
the existence of at least two dis-
Ising interaction
and
all
large
325
Quantum Spin Systems The
provide the starting point for a perturbation Heisenberg interaction. Now one has
estimates
Ising
of bounds for the
calculation
\\^^\P,U,)\\<e-^^-'\\X^^(ö)\\ with
;^A(^)
X^
In Order to bound
F(t) and note that uses
VyPy
we
^
^-ß(V^+ÖT^}p^^ß(VA+ÖT^+V,+ÖT,}
introduce the function F
=g-^(l-O^Ag-M^A+^^A)p^gm(^^A F(l)
=
X^(ö)
F(0)
and
=
Py
(0 (L)
-r
-Up
Tf'^(^)-^^-(^-
'^/^X^A) bounds
again
ößF(t) 4vV.(rA + r,; ^y) -öß,^ßn.(T^}F(t} i-ß(^^_^^^I/^J2^^iJ + -rUpr^LJL.ß^^_^^^2^
=
T^ denotes the local Ising interaction, evolution perturbed by Fy, i.e.,
T^'^'
x](A]
5^v))gAl-0(^A + f^v)
where the latter identification
where
then
+ ^/)+^(^A +
0. Moreover,
=
dF
But then
by
=
T^T^ where
=A unless ;c,
-^Tßt^y
where the
\\Tßt^^y\\
<
or
=
are
^'')^^^-(^-^-^^)^
e^^^^Ae'^^''^
y, is in
r^^,y
aj\y\e^^J^.
F(t)=F((^]-dß
i'^A]
+
e^^^^Ae~^^^^,
T^(^)
AI (7),
.
or
In
and T^''^ is the
.
particular,
AI (7). Thus
if yi
'^[^.^y]
G
T^'/(rA + r.)
localized around the contour y and
=
satisfy
Then
r^^{[4(i_,)(rA),F(.)]-F(.)r^(i_,),,}
.
J0
give a perturbation expansion X!^(d') =F(1) in terms of Py F(0). Convergence of the series can then established through term by term estimates following the arguments used This
integral equation
can
be iterated to =
of be to
prove Theorem 6.2.4. The estimates are uniform in A because of the localization properties of the Ising evolution and because the evolutes only
T^^(rA)
occur
in commutators with elements localized
calculation is outlined
äs
on or
around the contour 7. The
follows.
integral equation for F(t) is an n-fold integral, over a set 0 ^ < l, multiplied by a factor (öß^. The integrand /2 ^1 of which is formed by a mixture of commutators of each 2" contains terms with F(0) =Py and products with r^(i-/.),y Now Tß^^y is localized '^/^(i-r,)(^A) around -y in a set with at most (2^ -h l ) 1^1 points, the points in 7 and their nearest neighbors. Moreover, T^(7A) is a sum of two-body interactions translated under the Ising evolution and hence a sum of (2^'*'^ + l)-body interactions. iterate of the
The -th <
<
<
<
.
.
1^(7^)
Therefore the commutators of localized in
a
set with at most
commutator with
Tßt^y
|7|
has the
+
Py has at most (2^"^^ points. The number
-h
bound but is locahzed in
a
with
2^"^^
same
+ l
1)|7|
terms
of terms in
a
set with a
possible (2^+^ 4- 1)|7| points. Now one estimates the total number of terms in the th-integrand by the counting argument used to prove Theorem 6.2.4.
326
Models of Quantum Statistical Mechanics
(2^'+^ H- l)-body interaction and a local element with (2^+^ + l)h/| points. Therefore the number of terms is majorized by (2'"^-(|7| + n))" < n\ exp{2''+-(|7| + 77)}. But each term will have a norm bounded by ß'yg2/?(^, +...+?) x];ierefore one obtains norm bounds Effectively
one
Support in
||^,^(<5)||
a
has
a
set of
< l +
^(|.5|/?)" 7of dt,... Jo/''"' dt 2"(2*'+2(|y| + ))a/'e2/'('.+...+0
^ < l +
^(öl^le/'^)^'^
<
e^""^-\-'\(\-b\d\e^l^J]-^
n> l
for
a
suitable Z?
>
0
independent
of
dj
and
ß.
The last estimate is valid for
\8\< b-^e~^~f^J. Consequently, sup
(D^(P\py_)
<
xj' E Z''
3(l-/?|(5|e2^^')~^^/exp{-/(2jS7-2^'+2-log3)}<
1/4
/ >2
uniformly
in A, for all sufficiently large ß,ß> ß^, and ö satisfying \ö\ < ^-ig-2^7\ Hence under these conditions the anisotropic Heisenberg ferro-
magnet has at least two distinct extremal (T^,^)-KMS states. Therefore we have proved Theorem 6.2.48 for all (5 in a jö-dependent interval, ö G {-öß.öß}, and all large ß. This calculation, which is extremely, crude, has the unfortunate feature that ^^ -^ 0 äs ^ ^ oo. On the other hand it does have the fortunate feature that it does not
depend on the details of the foregoing calculation TA could be replaced by any Hamiltonian H^(A.) coming from an interaction ^ satisfying the assumptions of Theorem 6.2.4. The estimates only depend on the local transverse
part of the interaction. In the
structure of the
interaction and not its detailed form. In order to prove the füll one needs much more refined combinatorial arguments which take into account the details of the action of the Jf-7 part 7^ of the Heisenberg Hamiltonian. Statement of Theorem 6.2.48
One method of
improving the Peierls argument in the quantum framework adapt ideas of functional integration and to reformulate the v-dimensional quantum problem äs a (v + l)-dimensional classical problem. First we in-
is to
troduce
an orthonormal basis of the local vector space A which is in one-tocorrespondence with the set of configurations on A. It is convenient to label the vectors by subsets X C A and use Dirac notation \X) Specifically \X} is a simultaneous eigenfunction of the matrices {(73 ; jc G A} with eigenvalue
one
.
+ l
for the matrices
{0^3;
x
e
X}
and -l for the matrices
{
x
G
A\X};
the
scalar
product is denoted by (X\Y)(= öxj) and the matrix elements of ^ G ^A by (Jr|yi|7). Second, we reintroduce the spin raising and lowering operators cr^ (er] i(rl)/2 of Example 6.2.13. One has (cr^" 0 and =
=
[4,^5] with the
er
corresponding
follows that
(Tll 0)
=
=
T2^i,
[crl^cr^_]=^l
to different x,y
0 for all
X
G
e
A
A and
i^)-(n< xex
10)
,
mutually commuting.
It then
Quantum Spin Systems
327
Third, remark that the X-Y contribution to the Hamiltonian //$(A) of the anisotropic Heisenberg model is given by (5 TA where
E
T-A^-
7(^,>')K
+
=
It follows
E
-2
J(^^y}<^^-
+
^-<}
x,y G A
x,y e A
immediately
that
(X\T^\Y}=Q unless X and Y differ which
case
Trotter
by
point x being replaced by
one
the matrix element takes the value
product
lim
can
T,^^(^-/^//.(A)^)_
^IA-
.
/
\
traces. For
or
example,
^ {X,\A\X2}(X2\e-ß^-^^\X2}{X2\(I~ßöT^/n]\X,}
lim
''~^'^x,^^nc^
.
e
in ,
recall the
then be used with the aid of the orthonormal basis to
evaluate matrix elements
for each ^
neighbor y
2j(x,x''). Fourth,
(e-^^^^''(I-ßöT^/n)V
n-^CG
expression
nearest
formula
^-/^//.(A)_ This
a
But if ^4
both A and FA. Let
.
.
(Xn\e-ß^^'^^\Xn}(Xn\(I
-
ßöT^/n}\X,)
^^(X) Ylxex^3 ^^^^ 1^) ^^ ^^ eigenfunction of eigenvalue corresponding to FA. Then
=
denote the
E/^(X)
TT,,^(e-ß^-^^^c7,(X))=\im^^(X,\a,(X)\^^^^ X^t
'f{(X,\(I where
we
terpreted
identify Xn+\ =X\.
a
äs
figuration
x
is
ßdTf,/n)\X,^,}
representation the integer / E [l,] is inconfiguration at time t. Thus the of the classical configurations in subset äs a
In this
the time and Xt the classical
quantum configurations subset A
-
a
[!,
+
1]
are
on
sequence F
viewed
a
(v+ l)-dimensional lattice. Each quantum con + 1} of classical configurations {Xt\ ^ l, 2, =
.
.
.
,
changes the classical config governed by the value of the product in there But the representation. are a limited number of possibilities. Either in the factor and the Xt product is one, or Xt and Xt+i differ by one Xt^i nearest neighbor x' and then the factor in the point X being replaced by a Thus the is configurations of spins can be thought of äs product 2ßöj(x,x')/n. with the in time possibility at each step of a spin and one of its nearest evolving neighbors flipping over. Alternatively, if one adopts the particle picture, with the configuration X corresponding to particles occupying the sites x e X, then at each instant the configuration either remains unchanged or evolves by a particle jumping to a nearest neighbor site. Since we are examining the ferromagnetic model the weighting factor 2ßöj(x,x^)/n associated with the Jump from X to x' is positive whenever ^ > 0. In this case the quantum Gibbs state =X\. As the discrete time variable t
with Jr+i uration Xt evolves with the evolution
=
328
Models of Quantum Statistical Mechanics
defines
a
classical
sequence of
probability measures P over the (v + l)-dimensional F. If n(Y) denotes the number of jumps in the path and j(Y) the product of the j(x,x') along the path then the configuration F is given by n
configurations
associated with F
probability
of the
P,(F)
e-f^^^(^^f^(2ßdY^^^j(Y)l(Y.e-ß^^(^^'^ (2ßd/nY^^^j(Y))
=
where
^A(F)
=
Y.
^A(^r)
.
^r er
Note that the
probabiHty of a path of length 77, i.e., a path with n quantum Jumps weighting factor 3'^. So small values of d mean long paths are is the perturbation characteristic of the calculation. This unlikely. The probabilities for the Heisenberg model can now be estimated by combining the earlier considerations for the Ising model with an analysis of paths followed by the classical configurations. We will not give any further details of this analysis (see Notes and Remarks) which involves a variety of combinatorial counting arguments following the well-trodden path of random walks. Finally, however, we note that the conclusions for (5 > 0 can be extended to (5 < 0 by symmetry. Let LQ and L\ denote the sublattices defined by has
and Q
the
^'^((rl) a
=
(r\
a
=
{jc;
jc
G
1\ \x\
=
0 mod
2}
Zi
=
{x]x
e
1\ \x\
=
l mod
2}
*-automorphism ^Qi
X
LQ and
e
KMS State for the
State
lo
of ^ such that
O(.Q((T^)
=
cr^
for
Heisenberg interaction
for the interaction with -ö and vice
One
also
aQ(cr|) G
X
=
-cr|, aQ(ö2)
LI and
/
=
1,2,
with ö >Q then
=
-cr^ and
3. Then if
co o
a'^
is
a
co
is
KMS
versa.
analogous symmetry arguments to deduce a nonuniqueness Statement for the anti-ferromagnet from the ferromagnetic result. Let ao and a l be the *-automorphisms of ^ which correspond to spin reversal of atoms at points of LQ and L\, respectively. The assumption of nearest-neighbor interaction ^ in the ferromagnetic Ising model immediately gives can
use
ao(0({x,j;})) i.e.,
=
-^({x^y}}
=
ai()({z,y}))
,
ao and oc\ map the
interaction.
ferromagnetic interaction into the anti-ferromagnetic Consequently, in the absence of an external field there is an iso-
morphism between the sets of KMS states for the two models. This conclusion must be modified, however, in the presence of an external magnetic field because X
G
ao(/z(^)ö^)
=
h(x)(f^
with the
plus sign
if
G
jc
IQ, i-e., the sublattice spin reversal transforms
a
LI and the minus sign if
uniform external field into
staggered field. Indeed, one can show that the Ising anti-ferromagnet has at phases for low temperatures and a uniform, but weak, nonzero ex ternal field. This is not to be expected for the ferromagnet. Then the in-
a
least two
troduction
of
magnetization
a
uniform
and the
long
external
field
ränge order.
should
remove
the
spontaneous
329
Quantum Spin Systems
nonuniqueness result for the anti-ferromagnet with a translationally nearest-neighbor interaction and a uniform external field can be proved by a chain of arguments similar to those used for Theorem 6.2.48. The previous proof was based upon the observation that extremality of the (T^, ß}KMS State and invariance under the spin reversal symmetry are contradictory. The anti-ferromagnetic result is deduced from an inconsistency between ex tremality and translation invariance. Observation l is replaced by the following. The
invariant
Observation l'.
translationally
^ is
If
invariant
ü)(/^Pi)
sup
< i
It suffices to prove that
PROOF.
contrary, then
is
an
AceLo) >'-^i
x^LQ,y&L\
< l
,
states.
(T^, ^)-KMS
+
tr^)(t
+
a^))
inequalities
>(M-1)^
=
(M+
1)^
one
also finds
.
inconsistent. Hence
are
state. Assume the
/
\
P^_ and P^ in place of P\_ and P+ l
But these two
extremal
((l
lim
-
\x-y\-^oo
^) Using co(o^)-
a
in space and hence
\x-y\^oo
=
cü(PiP^)
sup
(i^^ß)-KMS
is not
co
strongly clustering
4co(^P^)
lim
l >
where M
o)
co
satisfies
x^LQ,y^L\
distinct extremal
two
are
which
,
x^LQ,y^L\
then there
invariant interaction and
translationally
a
(i^ ^ß]-KMS state
co
cannot be extremal.
Next we must adapt the definition of configurations and contours etc., to the anti-ferromagnetic case. Let ;c G A n LO and y ^ A r^ L\. We now examine + (or c(jc) ). The definition configurations c such that c(x) c(y) c(y) of a contour remains unchanged but the class of contours r(c) is taken äs the =
=
of 7
set X
e
=
LQ and
{(^1, 3^1), y e
LI,
.
.
we
.
,
äs
which separates x and y but the connected set such that x G z
G
AC
n
Z/i. Thus
Observation 2'.
one
with
(xn^yn)}
argue
c(xf)
=
c(yi}.
Since
c(x)
before that there must exist
=
c(y),
and
y(c) set A^ C \x(y(c}) is now the largest NN(A)for -{- for z G A^ n LQ, c(z) A^, c(z) a
contour
=
obtains:
For any
region A
one
has
n^i<E^^' y
\\^here
^r=
n ^c(s]P'c(t] (s,t}ey
(and c(s)
=
c(t) by definition)
The enumeration of the contours y remains the replaced by the following:
same
and Observation 3 is
330
Models of Quantum Statistical Mechanics
Observation 3'
If x
.
^
LQ^
y ^
LI and
COA(P.)
for large
A
andfor
some
K >
co\ is
a
state
^A such that
on
e-^\'\
<
then
log 3, oo
coA(PiPi)
<
X] /
Finally,
if COA is
a
(T^,/?)-KMS COA(P.,)
==
n^-'e-^^
< +00
.
2v
state,
one
has the bound
\\^fß(P,U,)\\
<
It remains to make
To
reconsider the
different
Let
an appropriate choice of Uy. ferromagnetic case from a slightly
explain this angle.
choice let
us
\X}
denote the orthonormal basis of the local vector space A i-ised are in one-to-one correspondence with configurations cx of and the effect of the ferromagnetic Uy is such that
above. Then the X
spins
U,\X} where cx^ is the
reflection
=
unique configuration
\X*}
with
(*)
r(c;^*)
the contour y. But the \X} part FA of the Hamiltonian and the calculation
rephrased
removes
r(cx)\y, i.e.,
eigenfunctions following Observation
the
of the 3
spin Ising
can
be
äs
T^ß(PyUy) where
=
are
E/^(X)
=
P,U,e-ß(^^(^^-^-(^'^^
(**)
is the
eigenvalue of FA. proceed in the anti-ferromagnetic case. One chooses with the unitary Uy property (*) but with the anti-ferromagnetic definition
This indicates how to
any of r(c). It then follows that
(**)
remains vahd and
one
argues almost
äs
before. The
translation-invariant, nearest-neighbor anti-ferromagnetic Ising
inter-
action with external field has
<1.(W) for all
X
e
=
-ha^
Z^
^({x,y})=jalay, with 7 > 0 if
|x
-
yl
=
l and
^(X)
=
0 in all other
E^(X)-E^(X*)
>
cases.
Therefore,
2(7 -H) M
and
\\r^(p u}\\ \\i-ß\^rjUy)\\ for
^
>
0. Let
co
interaction. Either one
KMS state
foregoing
< ^
e-^~ß(j-\^\^\y\ e
be any limit point of the local Gibbs states COA for this co is not Z^'-invariant and there is automatically more than
or
it is
translationally
invariant and
estimates the conditions of Observation V
by combination of the valid for ß sufficiently
are
331
Quantum Spin Systems
Ising anti-ferromagnet has at least sufficiently large ß if the external field satisfies \h\ <j. But this conclusion can then be extended to the anisotropic Heisenberg model by the perturbation calculation outlined in the ferromaglarge. Therefore
conclude that the
we can
two distinct extremal KM S states for
netic
case.
interaction mth
Heisenberg uniform external field h. Itfollows that there is
netic
exist
öe
translationally invariant anti-ferromagnearest-neighbor couplingj, anisotropy 6 and
Let O denote the
Theorem 6.2.49.
least
at
two
Assume
>
v
2.
>
Q andfor each
extremal
(i^^ß)-KMS
ß^
a
ß
>
ß^, aöß > 0 such that for all ß > ßc
states
there <^nd
{-öß.öß}.
The value of öß obtained
exponentially
fast
ß
äs
mates is better. One
^
can
by the
crude
perturbation
establish bounds with
whether the model exhibits
calculation tends to
The refined calculation based
oo.
O(ß~^^^].
öß
anti-ferromagnetism
for all ^
path
zero
esti-
It is not known
(1, 1)
G
several indications that this is the case, e.g., the classical and numerical calculations. are
the
on
but there
Heisenberg
model
Heisenberg model appears to be somewhat isotropic case. In two dimensions it is possible to l, i.e., rule out the possibility of spontaneous magnetization for the ferromagnetic model, or spontaneous magnetization of the sublattices in the anti-ferromagnetic model. Although this does not entirely exclude the possibility of nonuniqueness for the corresponding (T^, jS)-KMS states it does rule out the most likely mechanism. Instead of establishing this result directly we derive a more general Statement on the absence of spontaneous symmetry breaking based on the general criterion in Theorem 5.3.33A. The properties of the Heisenberg model will then follow äs a corollary. The
structure for the
phase
different \i d
in the
=
Theorem 6.2.50 group, g termines
V(g)
K^
an
(Fröhlich-Pfister). a
unitary representation of g G GH->
action
Let G be
yx(9]
Ad(F(ör))
compact, connected Lie
a
G
^^
^^^^
^^^ ^P^^ algebra ^ over under the action of G, which invariant is interaction
action
y(g]
=
xei^yx(9]
C^"^^
on
^^
j}
.
~
9)^^-^
^{x} Let
which de-
^^d hence the
^
be
a
two-body
y(g](^(X))^<^(X] all g ^G and all
for
two-point
sets
m{x,y})\\ for
some
Ifi^ State
for
C(\x-y\
+
ir'
C >0.
denotes the evolution associated mth O and some
ß
^
if
CD
is
a
(i^^ß)-KMS
R then (D o
for
<
X. Further suppose that
all g ^ G, i.e., there is
no
y(g)
=
(D
spontaneous breaking of symmetry.
332
Models of
Remark. are
One
in many
PROOF. L=
Quantum Statistical Mechanics demonstrate
can
senses
by example that the conditions optimal (see Notes and Remarks).
Fix g ^ G.
By Theorem 5.3.33A it suffices 1,2..., such that
y(g}(A) for all ^ G ^ and, in
=
find
to
a
in this theorem
sequence
UL
e
2.1,
lim^ULAUl
addition,
SUp||^cI>(^i)||
<(X)
L> l
where ^(D denotes the generator of T^^. The first attempt to construct such would be
UL where A^
a
sequence
^^,^^,V(g)
=
{(x\,X2) G Z-; x/
Choose the
path
t
one-parameter group
a -^
g^
over
0
t
-^
g^
of G such that
that g^^ V (g) is faithful and then G is Now define
t/L where
g^
=
for
g
x
/c-2L-max{|.Ti|,|x2|}.
AL
G
=
a
closed
and
g^f^
g^
for
(.X, y)
it is convenient to X-
on
Z".
;c
==
(;ci,;c2)
~
with
=max{|.Ti ->'i|,
y
\\Ul^({x,y})U,~^({x,y}]\\
=
||F'(0,)
=
K* (0,)
II
.^2-
3^2!}
|.T|
> 2L
and
y\
terms with
V^(gy) <2||Ffe)-F(3v)ll
<2n\\x\-\y\\
Y^ L < or
\x\
||
.
2n\\x\-\y\\-\\<^({x,y})\\
< 2L
L<\ v\
<
2L
< L
and
For the
^(3,) O({^, v})|| V(9y) r(gy)({x,y})y{g,) F(g,)ll l|1>({.Y,y})||
Therefore <
\x\
> 2L are zero.
r(ö,)({.T, v})F(<7.v) r(ö)({-r,>'})F(a.) '
.
the metric
use
the terms with
'})||
||<5o(t/i)||
A2i\Ai
6 Z-
Then, using the invariance of <E)({.T, y}] under y(g], the
\y
\\Ul({x,y})U,
G
Then
^ C z-
sum
g and that the length of possible since we may assume connected subgroup of U(N + l). =
(8).reA2, V(gx)
\\d.,(U,]\\
g^
^ is at most ntQ. This is
< ^ <
-
333
Quantum Spin Systems Hence
11^0(^/1)11
X^ 2n\\x\-\y\\C(\x-y\ + ir'<^
<
(^.y]^^~
since the
sum
is
over
two dimensions. Thus Theorem 6.2.50 follows from Theorem
5.3.33A.
implications for the Heisenberg model. 5(9(3) of rotations acts äs a continuous group spin-^ quantum algebra ^ by rotation of the spin on a local algebra ^2lA is deterrained by unitaries R^. But the interaction of the two-body Heisenberg
This theorem has several
First, remark that the group of *-automorphisms of the vectors cr^. The action 7
with
exp{z J^.^^^fl.cr^} model, 6({z, j}) of the
Spin
ticular
G
vectors a^ and
external field and KMS State
a
no
(I)^^^({jc})
those with
are
no
(t^,ß)-
60^(0^)
expects
>
0.
energetically favored spins positively aligned parallel to the third axis. Then at sufficiently low temperature any (T*,jS)-
KMS State obtained in the limit
ism, i.e.,
deduces that each
one
=
configurations one
interactions
many-body
model with external field
Thus
O is invariant under y. Hence if there is
cr^', i.e.,
of the one-, or two-, dimensional model is y-invariant. In par0 for alljc. But if co/^ is a (T^, jß)-KMS state of the ferromagnetic
co
co(cr^)
is invariant under the simultaneous rotation
j(x,y]o^.(jy,
=
one
h(x)(j\
=
\\h\\^
>
then the
0 should exhibit
a
residual magnet-
should have lim
lim
lA^r^ V co;,(cr^)
|,.,U_Ol->ool
0
>
^Z..^
.
But any such limit state is a (T^,^)-KMS state for the limiting dynamics, i.e., the interaction with h (},\^y Theorem 5.3.25. Thus Theorem 6.2.50 demonstrates that
senberg
spontaneous magnetization
model in one,
or
cannot
for the
occur
isotropic
Hei
two, dimensions whenever
sup^ x-y\^\j(x,y]\
.
X
y
if
Alternatively,
{d(o\G\ + 020-^)
+
of rotations of the a
(T*, j8)-KMS
one
considers the
0^0-3} spins
model
anisotropic
can
=
j(^,y]'
around the third axis. Hence if co/,
now
subgroup corresponds to
state with external field in the first direction
l,l,o.l^l^^l''^5^""('^) and there
^({x,y}]
then the interaction is invariant under the
be
no
residual
magnetization
=
'
in the first direction in either
one or
two space dimensions.
also prove the absence of spontaneous magnetization by use of the Bogoliubov inequality mentioned in the Notes and Remarks to Section 5.3.1 äs
One
an
can
early form of
a
correlation
inequality. Lei
Lemma 6.2.51
(Bogoliubov Inequality).
the C"" -algebra
2l and let d denote the generator
co
be
a
of i.
(i:^ß]-KMS
state over
334
Models of
Quantum Statistical Mechanics
Itfollows that -iß(ß(AA' +A'-A)co([C,d(C]])
>
2|a;([C,/l*])|-
forallA e'^andC eD(ö}. PROOF.
Let
r^
l
/
(A,B)^=-
^lco(^*Ta(^))
ßjQ
denote the sesquilinear form orem
5.3.17. The
on the analytic elements ^l^ introduced prior Cauchy-Schwarz inequality gives
\(A,B)^\~<(A,A}^(B,B)^ EiiiifB
=
.
ö(C)
(A,B)^=-
f
P
Jo
=
=
by
to The
dX-^co(A^,a(C)) äA
l
(iß]-\co(A'iiß(C]) (ißr^o,([c,A^-]]
co(A^C]}
-
the
(T,^)-KMS condition. Alternatively, if U(o(t] exp{zY//f,j} representation (,,^, TT,,;, Q.,,,], (A,B]^
=
< =
unitary
group which
implements
(7i,,(^)0.(^//,,)-'(l -^-^^^'0^o.(5)a.)
Therefore, the inequality (l
(A,A)^
is the
e~''^)/x
+
Combining this estimate with inequality for (A,B)^, one finds
in the
-
(l +e~^)/2 gives
<
(^-^e'^^-)7ia,(A)Ü,)/2
(n,,(A)ü,,
{aj(A'A)
T
co(^*T,/K^))}/2
the
previous
=
co({A\A})/2
.
evaluation and the
Cauchy-Schwarz
r'K[c,^*])|-^|(^,5L|
-(/^)-^co(K,^})co([C,r])/2 or
-ißo4{A-^,A}}o3([C,ö(C')]) for
T-analytic
A and C. The füll result follows
in ^l and it is also
>
2co([C,^*])|-
by continuity
because
^I^ is
norm
dense
for ö.
a core
We next demonstrate how the
reproduce the earlier conclusion
Bogoliiibov inequality
can
be iised to
the absence of spontaneous magnetization for the two-dimensional anisotropic Heisenberg model. Assume there is an external magnetic field ^({x}} -h(x}(f\, in the direction of the first axis, and set \\h\\ I^WI- Further assume there is on
=
=
a
siip^^^.-
two-body interaction.
({x, y})
=
-j(x,
y){<5(cr{<
+
')
+
o^<,^}
Quantum Spin Systems
335
and set
^ ^-y\^W.y]\
sup
=
Z^ '
:f
^2 yef ^
We aim to show that there is
lUII
<
Let
no
spontaneous magnetization whenever
=-
Afl denote the lattice of pairs (pi-, pi) where
integers
such that
fl//2
<
< /
af
fl//2.
^ e-'>x
=
2nni/ai
pi
Introduce the Fourier =
and the HI
transforms,
are
a/,
by
(^ro*,
X^^a
where px
=
p\x\ + /72-^2. Thus
<
IA.I-' ^ e'>^af
=
.
peA
Next define the
mean
M,(/z)
=
magnetization Ma(h] by
|A,r^ Y. Z^ ^(^i)
=
iA.r'^(^?)
Afl
jce
and introduce
M(/z) We will bound M (h]
equality.
=
lim sup IM
(Ä) l
.
by choosing A d2^C d^^ in the Bogoliubov the observation that by =
=
in-
This choice is motivated
|Ar>([a3'',a-''])P=4M(Ä)2 Moreover,
one
.
has
|A,-^5]co(afa-'' + a2-X)/2=l
-
/?eA
The idea for
determining
the bound is to find
a
volume-independent positive
continuous function E such that
-ißa)([dl5(d-'')]) The
<
Bogoliubov inequality then implies
\A,\E(p)
.
the bounds
\A,r^^co(a^,a-^ + d-''a^)/2 ^2.
1=
2
peA
|Aa
>
"E
j
iH[a^,.a-/])\
2
\ßco(laP,,S(a-'')])j
pe^a >
4M,(Ä)2|A,r' ^(p)-'. P^^a
Thus, taking the limit
M(hY
a
<
^
oo
and
4(2n}
rearr
-2
anging gives
d^pE(p]\Pi\< \P2\<
336
Models of Quantum Statistical Mechanics
To find
an
appropriate E one has
-ißcD([dld(d-p]]]
Y,
^ß
=
^-S
calculate
a;([(7^, ^(0-3'^)])
7(^,j^)(i-cosX^-y))^K<
and this
^M)
+
E E J(^^ y}^(<^\^l + ^l<^2} +^ßY. -r
e
gives
Aß
x,y e
+
to
Afl
_y 6
A^
.X
E
h(x)o,(
.
A
Hence
\im&u^-iß\^,^^(J,([äl^(a-'']])<m\\}\\p-+m\\h\\ a>oo
Combination of these results
M(hY
<
4(271)-^
yields
the bound
(^^^^^d^p(m\\j\\p^+m\\h\\r \P2\<^
An alternative weaker bound
be obtained
can
by integrating
over
the
^ TI^- This, however, allows the introduction of radial coordinates and hence one obtains smaller
region \p\\^
M(h]^<
+
\P2\~
\4(2n)-^ r^/';^(8|j8|||7||/+4|/J|||/;||)-'lJ [(8|;ß||U||7r)-'log(l+2rr2||y||)/||Ä||)]''.
L =
Hence
M(h)
-^
0
äs
Jo
\\h\\
0 and there is
-^
no
residual
magnetization
in the
direction of the first axis or, by symmetry, in the plane defined by the first two axes. A similar argument establishes that the A^-7model has no spontaneous
magnetization in one or two dimensions. A slight modification of the proof rules out the possibility of spontaneous magnetization of a sublattice for the general Heisenberg antiferromagnet. One a^ by A replaces A a^^^ and considers the generalized magnetization =
=
M,(h]
=
|A,
-'
^
-iPx
co(cr{)euj^u^je
xe^a
Despite these
results
one
should
is rather weak. One has M (h)
this bound
diverge
at the
^
perhaps emphasize \ log ||/z||p ^" äs \\h\\
that the bound >
on
M(h)
0 and all derivatives of
origin. Thus, the argument does magnetization,
not rule out
singular
behavior of the derivatives of the The above argument existence of
can
multiple phases.
in certain
cases
be inverted and used to prove the
The estimate
-i(o([a^,,ö(a-P)])
to
a
bound
on
the energy of excitations around
equi-
librium. The behavior of this energy at zero momentum, i.e., at /> 0, was the critical feature that ruled out ferromagnetism. The basic idea for exploiting =
infrared bounds to obtain the
opposite conclusion can Suppose that co is a Z^'-invariant (T*,^)-KMS state and some ß ^ U. Next, introduce the functions FA by
be outlined for
some
äs
follows.
interaction
Quantum Spin Systems
FA(X) where
Ä
=A
-
automatically
co(y4)1l
As FA is
.
(D(Ä^^,(Ä)},
=
Ae^
function of
a
the Fourier transform of
FA(X)^
l
,
positive type
a measure
337
of the lattice Z^' it is
fi^ ?
d^,(p]e^P^
.
J\p\<-K If
is not
strongly clustering in space, FA will have a nonzero asymptote, for ^, and this will be reflected by the presence of a Dirac point measure (5 at ;? 0. Thus, to deduce that co is not extremal (T*^, j5)-KMS it suffices to find an A for which this point measure occurs. Next, we remark that HA should be related to the dynamical propagator, or Green's function, of the Hamiltonian determining the excitations around the equilibrium state. The variable p corresponds to a lattice momentum and the energy of excitations should be given by a positive function E of p. For p ^^ general dynamical reasoning indicates that /x^(/?) (ßE(p]]~^ The factor ß enters because the inverse temperature determines the scale of energy. Thus, one expects co
some
^ G =
~
.
dtiAp)=c^d(p) for c^
>
0 and
J^
>
0. But
+
has the
one
d^(ßE(p])-^d^'p
simple identity
(o(A*Ä)^F^(0) =
l
df^Ap}
J\p\
=
c^+^ J\p\ [ P
d^pE(pr' <
71
Now suppose
/
d^'pE(p]-^
< H-oo
.
J\P\<71 In this
case one
term in this
must have c^ > 0 for
identity
argument is
for
co(/l*J^)
sufficiently large ß because the second zero äs ^ co. Thus, the crux of this
tends to
>
estimate of the form
an
^A(p)<ßE(pr' E such that E~^ is integrable. application of this technique to the anisotropic Heisenberg model with translationally invariant ferromagnetic interaction. One expects a spontaneous magnetization at low temperature. Hence, any spin reversal Symmetrie KMS state co should not be extremal and the above argu ment should apply to
for
some
Let
^ G M and
US
an
examine the
F,3W=co(ff
model is
Heisenberg two-body interactions (see model
one
integrable
the
the presence of a
lattice
.
analogue of
a
continuous System with 6.2.1). Thus for this
Notes and Remarks to Section
expects the low
at
a
momentum relation
for all
E(p)
cp^.
Now
E(p}~^
is
3 and hence this intuitive argument indicates origin in transition all dimensions greater than or equal to three. phase v
>
338
Models of
Quantum Statistical Mechanics
this method of infrared bounds is
Although
seductively simple
is nonetheless difficult to
apply. The necessary estimates derived. Some results have been obtained through bounds Gibbs States (DA which
are
uniform in A. In
on
on
in
outline, it
easily JLI^ the finite-volume are
not
the method has been
particular,
used to prove nonuniqueness of (T^, ß)-KMS states for a variety of classical models, for the isotropic Heisenberg antiferromagnet, and the X-Y
successfully model for
spin greater
than one,
i.e.,
A/' + l > 3 and
We conclude this subsection with
tremal KMS states. The
a
v
few comments
>
3.
on
multiplicity
of
ex-
model for which much information is known is
only translationally invariant Ising model. In this model it is known that there is one, and only one, extremal (T^, j5)-KMS state for all positive ß if the external field is nonzero or for small positive ß if there is no external üeld. Furthermore in the absence of a field there are exactly two extremal (T^^, jS)-KMS states for ß > ß^ where ß^ is the maximum value of ß for which uniqueness occurs. These two states are clearly translationally invariant. In three or more dimensions the Situation for the Ising model is known to be more complicated. There is still a unique phase at high temperature or in the the two-dimensional
presence of
number of
an
external field but at low temperatures there can be an infinite which are not necessarily Z^'-invariant. This can be partially
phases
understood in terms of
juxtaposition
in the third direction of the two two-
dimensional pure phases. Similar easily understood effects occur already for the ground states of the one-dimensional Ising model, äs we shall see in the next
section.
6.2.7. Ground States Ground states,
analyzed
in
zero-temperature states, were introduced and partially Chapter 5, see Definition 5.3.18 and Proposition 5.3.19. These or
states can be described in various ways, but
physically they correspond to the a given dynamical group T. Ground states positive are sometimes referred to äs (T, +oo)-KMS states, a notation which is partially justified by Proposition 5.3.23. This proposition establishes that the zerotemperature limits of KMS states are ground states. Nevertheless, the set of T ground states has afiine properties dififerent from the set of (T,^)-KMS states. The set Ä^oo of ground states is always a face in E^i^ but it is a simplex if and only if (^,co) is R-abelian for all co G K^, see Corollary 5.3.40. The set Kß of KMS states at finite ßisa simplex, but it is a face in E^ if, and only if, ncoC^} is abelian for all CD G Kß (see Remark 3 after Theorem 5.3.32). In particular, spin Systems provide some striking examples of the differences between ground states and finite- temperature equilibrium states. Consider the one-dimensional quantum spin Systems with translationally invariant nearest-neighbor ferromagnetic Ising interaction
339
Quantum Spin Systems
external field the same remains true for the T^ ground states. If the field is in the positive direction, the ground state is determined by the configuration in which all atomic spins are positively oriented. If the field is in the negative direction, all spins are negatively oriented. But if the external field is zero, a drastic difference occurs. First, there are two extremal Z-invariant T^ ground states, the states with all atoms positively, or negatively, oriented. Second, there are an infinite number of nontranslationally invariant ground states. A typical such State cox is determined by positive orientation of all atoms to the left of jc G Z and negative orientation ofthose on the right.^ Thus, the multipHcity and invariance properties of the T*^ ground states and the (i^,ß)-KMS states, even with ß very large, can be quite different. Furthermore, the states C0;c,jc G Z, are all extremal T* ground states which are mutually unitarily equivalent. Extremality follows from purity, and the equivalence arises because one state is transformed into the other by reversal of a finite number of atomic spins (cf. Corollary 5.3.40). But for j? G IR, extremal (T^, jS)-KMS states are either equal or disjoint, by Theorem 5.3.30, and hence the equivalence is another distinctive feature of zero temperature. Finally, an explicit calculation shows that the finite- temperature KM S states cDß of the Ising model converge in the weak* oo but the limit state CD is not extremal even among the Ztopology äs ß invariant ground states. It is the average of the spin up and the spin down ground states. Thus, purity properties do not necessarily survive the zero-^
temperature limit.
Despite
the differences illustrated
similarities between to
stability.
ground
In Section 5.4
by
the
Ising
states and KMS we
interaction there
are
many
states, in particular with respect
established various forms of
stability
for both
types of state and we conclude this discussion of spin Systems by showing that the ground states can be characterized by stability properties analogous to the maximum
entropy principles of Sections 6.2.3 and 6.2.4. This type principle was expressed äs maximality of a functional S ßH^ or, equivalently, a functional ß~^S-H^. Thus, the zero-temperature, jö 4-00, analogue corresponds to maximization of //CD, i.e., minimization of the energy functional //$. We begin with a zero temperature version of Theorem of
=
6.2.36.
Theorem 6.2.52.
corresponding T^
(1)
Let O be
local
an
interaction
automorphism
converges
strongly
the
group. Assume that:
to an
lim
of a quantum spin System andi^
automorphism
group T, i.e.,
||Tf(^)-T,(^)||=0
A->00
(2)
for allAe^i and t G (R. The surface energies W^(I^)
are
\vell-defined
elements
of
^
for
all
ACL. ^
It is not
the result
immediately
we are
about to
evident that the co.x
derive, Theorem
are
ground states but this Example 6.2.56.
6.2.52. See
is
an
easy consequence of
340
Models of Quantum Statistical Mechanics
(3)
Ö^U^A ACL
is a core for the generator 8 of i, The following conditions are eqiävalent for
(1)
CO(ÄD(A))
a state CD.
CD'(H^(A))
inf
-
co'ec-
all A c
for
I, where H^(A)
//o(A)
=
+
^^(A)
is the conditional
energy and
C^ (2)
CD
is
{co';
=
groiind
a i:
co'
e
21,
'lat,.
state, i.e.,
-icD(A'' d(A)) for
all A
cD(ff^(A)}
0
CD(T(ff^(A)}}
<
all \ C L and all positive maps T
T (A) =A^oi Ae
Remark.
>
eD(ö).
(3)
for
kJ
=
The remark
:
^
^
M vi'ith the proper ty that
^^c.
to Theorem 6.2.4 is
prior
again relevant; an interaction 3^' d(= d^) and the W^'(\) are well defined. Thus Assumption (2) is partially redundant. Moreover, the results of Section 6.2.1 give various explicit conditions on <|) such that the assumptions of the theorem are valid, e.g., '
can
be chosen such that
always
ii*iL
=
=
E^'V^p E V-^^^,,^;,
II* W II) /
^0
for
some
/l
PROOF.
(1) A e
^
>
0.
We will prove (3) : If r : ^r
^A^- then co'
(3)
=>
(2)
:
=
co
o
This is
(1) ^
^
(3)
^ is
T is
a
a
(2)
(1).
=^
positive
map with the property
state in the class
proved in
a
C^^
and hence
similar fashion to the
Theorem 6.2.36. If A
=
A* G
^IA, then condition (1) implies that
co(e''-^Äp(A)e-'^^) and
by diflferentiation
at z^
==
0
one
a core
for
(5,
the state
co
>
co(H^(A))
deduces that
co(ö(A)) Since D is
<+~
=
0
.
must be i-invariant.
T(A]
(3) follows implication (1)
=A
for
from
(1).
=^
(2)
of
341
Quantum Spin Systems Now reconsider the operator 7^; ^i-^5I introduced
yB(A)=B''AB-{B''B,A}/2 It follows from the argument
prior
5.3.15,
.
to Theorem 5.3.15 that
prior
to Theorem
of positive maps. Furthermore, ifB e ^A then hence Tt(A) A. Condition (3) then implies that
semigroup
t\-^Tt
75(^)
=
is
exp{tyß}
0 for^ e
=
a
^A^ and
=
co(r,(^ci>(A))) for all ^
>
co(fi^W)
0. Therefore
>
ü;(7ß(Äa,(A))) But the T-invariance of
implies,
co
äs
in the
-;w(5*5(5))
=
0
>
.
proof of
Theorem 6.2.36, that
co(ys(ff^(A)))
and hence
-ico(B*ö(B)) Since Z> is
for
it follows that
>
0
.
a T ground state. co ö, (2) =^ (1): As there is no analogue of the Gibbs condition for ground states this proof is quite different to the proof of the corresponding implication in Theorem
a core
is
6.2.36.
We first need
selfadjoint
a
result
extensions of normal states to semi-bounded
on
elements affiliated with
lemma is formulated for
a
Neumann
von
algebra. The following readily to semi-
but it extends
positive operators
bounded operators. Lemma 6.2.53.
Let W. be
a von
Neumann
algebra
positive, selfadjoint operator affiliated mth 50l,
{^n}n>i
^
sequence
of vectors
The following conditions
(1)
sup{a;(J5);
(2)
^
B G
are
äR+,
eD(A^I'^]for
Hubert space
normal
state on
9),
A
a
9Jl, and
such that
in
EII^"|'
ö^-E^^-.' 77
on a
co a
> l
=
I
n>\
equivalent: 5 <
^}
< + cx),
allnand
5]M^/^^J|^<-foo
.
n>\
If these conditions
fulfilled,
are
then
sup{co(5); Bem+,B
=
Y. \\^"~^n\\^
'
n>\
Remark.
Recall from Theorem 2.4.21 that if
existence of
a
sequence
{^n}n>\
co
is
^^^^ ^^^ assumed
a
normal state
properties
on
9J1, the
is automatic.
342
Models of
PROOF.
Quantum Statistical Mechanics
Define
K^
sup{cü(5);
-
9Ji+, B
B e
,
^2-5:lMV2^^Jl^ n>\
whenever, the right-hand sides exist and define K\ and K2 Let /,j be the sequence of functions over [R+ given by (
.
,
,.
ifO<x
X
^"W-|, and define Af,
by A^
,f
^>^
/(^)-
=
It follows from Lemma 2.5.8 that
theory
fn(A)
e
931 for
n
=
it follows that the
the form
a
quadratic forms defined by A,, defined by A, i.e., if (^ G
A]^a(]-l^^'''^'^'''^^ a(^)
(f A^) (^,
to be +00 otherwise.
^
-
^
+00
1,2,
.
.
and by spectral monotonically to
.
,
converge
if^eö(^'/2) if^^D(A^r-)
,
.
Therefore
V(4,^^,)- E(^'^'^^-'^'^'^^-)
lim
"^^4l whenever the
^/^
e
D(A^^-).
r^i
(See the discussion of quadratic forms prior
to Lemma
5.2.13.) Hence K2
if 5 6
conversely
931+
w(B)
and 5 < ^ in the
=
^(^,,
54)
<
/c > l
.
sense
of
quadratic forms,
X](^*-)
=
then
^^2
k>\
Hence
^1
This resiilt indicates
a
<
A:2
.
canonical way of extending normal states to semia von Neumann algebra.
bounded operators affiliated with
Definition 6.2.54.
Adopt
the
extension of the normal state
9Ji is defined
to the
and notation of Lemma 6.2.53. The
semi-bounded operator A affiliated with
by
(o(A) Hence if
assumption co
o}(A)
< +00 one
sup{a)(5);
=
B G
9JI+,
B <
A}
.
has
a;(^)
=
^|M'/^^||-
,
n>\
where
Next
{^n}n>\
we
need
^^^
a
^^^
representatives of
result which
co.
implies that
states in
C^
are
quasi-equivalent.
Quantum Spin Systems Lemma 6.2.55.
Let ^be
C*-algebra of the form
a
^-M(^^o where M
algebra.
is the füll
complex
n
matrix
x n
Let (D\ and 0)2 be states
It follows that (D\ and 0)2
The
proof
algebra
a
and 2Io is
an
arbitrary
C*-
.
quasi-equivalent.
are
relies upon
,
M such that
on
<^lko ^^2^0
PROOF.
343
similar to that used to deduce Proposi
technique
tion 2.4.27. Let (/, TT/, Q/) (,7r,n)by
be the
cyclic representations assodated
S=-Ö12) Assume, ad absurdum, that
Q
71^7110712,
with o)/,
z
1,2 and define
Qi0n2-
=
and 712
are not quasi-equivalent. It follows readily representations, which we may take to be TI\ contains a representation which is disjoint from 712. If P is the smallest projection in Ti\(^]" r\ Ti\(^]' containing the projection onto this subrepresentation, it follows that the projection P00 on S commutes with 7r(3I)'. Öi2 Hence, P00 e 7z;(^)''n7c(^)'. The von Neumann density theorem, Corollary 2.4.15, now implies that there exists a net A^^^ such that
from Theorem 2.4.26 that
7i\
of the
one
,
=
n(A^] in the strong
topology.
But this 711
(A)
^
that
means
^
P 0 0
712(^)^0
P,
,
and hence
C0i(^) The
if
inequality
{^/7}i
->
follows since P are
(Ql,Pni) ^ 0, G 7ri
(^]" n
matrix units of
n\
W2(^a)
(^)',
P
^
^ 0,
0
.
and
QI is cyclic for
TI\
.
But
M, n
A^^^
E(A]
=
"^Eii AEii i
defines
=
\
a projection from 51 onto 5Io M^ u 51. (For a similar argument see the proof of Proposition 2.6.9.) As 711 (^a) and 7C2(^a) converge to the elements P and 0 =
of the Centers of 711 and 712, it follows that n
lim
Tii(E(A^]]
=
lim "^
j^\
(
-
V 7r/(^yi)7i/(^a)7r/(^iy) "
(
^ ni(Ejj]
\7= l
But
äs
E(AO^)
e
^0,
we
the limit
over a
\ lim7r/(^)
\imTii(A^)
-
/
have
coi(E(A,)} Taking
^
=
CD2(E(A,))
leads to the contradiction
.
=
Ö^P
.
344
Models of
Quantum Statistical Mechanics
(a,,pn,) Therefore 7i\ and 712
is
a
state and let
ground
o
.
quasi-equivalent.
are
ENDOF PROOFOF THEOREM 6.2.52. O)
=
CD
It remains to establish
(2)
^
(1).
also denote its normal extension to 5PIoj
Assume that
T^(o(^T
=
The
and generator H^j of the canonical unitary group Uco which implements T on j leaves Q^J invariant is positive and HO, is affiliated with 9}?^,^ by Proposition 5.3.19. Now if
i/^
G
D(H,,) ilH^o
for A G ^IA and
and A G
-
i//
G
then
D(ö),
7i,,(^)i//
7r.(ÄD(A)), 7i,,(A)]il^ /)(//,o).
Thus if
D(H,,). Therefore,
G
=
(7i,M^})
=
Q
U^(t)
=
-
Qxp{it(H,,
7i,(ö(A)))il,
7r^(ÄD(A)))},
-
one
has
U^(t)n,(A}U^(t}-^i^ =
7i,,(A)i^
/
+ /
dsU^(s)[H,
7r,,(Äi,(A)), 7l,,(^)]^AW"^A
-
Jo =
7Coj(^)'A
for all ^ G ^A and the Trotter
i// G /)(//ro). Consequently, Uj^(t) product formula,
t/A(0 that
^AW
e
affiliated with
proof of Proposition
Thus, ^yA n
7ioj(3I)".
2.6.9
imply
Tr^l^U)'
^
Tü^X^lf
n
-
7r,o(ÄD(A))
-
7r,,(ÄD(A))
is
that
is affiliated with
Lemma 6.2.55 that co' extends to
and //,,
But Lemma 2.6.8 and the argument used in the
TIoX^A/ ^^03W-7Ü,,(^AO'' i.e., H,,
But if follows from
lmjU,,(t/n)Qxp{-itn,,,(H,^(A))/n}Y
-
7r,,(^l)^ 7ioj(^A)'
7r,,(^lA)'.
G
a
Tr^X^UO"-
,
Now if d G
normal state of "JR^j and
C'^, d(R,,,}
it follows from may be defined.
But
^^L^^iUO" by continuity
Äo(A))
-
the monotonic definition of
CL)'(//,, Consequently, implies that
-
the
-
^>.('^^^}"
CD' (H,,
-co(//ci>(A))
-
the second characterization of
positivity
of H^o 0
=
co'(H,]
~
(*)
//ci>(A))
-
7if,j(ÄD(A)).
H^o
CD on
ÄD(A))
to'(Äi,(A)) Finally
^
and hence
co(H, by
^
Thus, +
<
00
.
CL)'(//(,J) provided by
co'(H,,,
-
ÄD(A))
Lemma 6.2.53
.
yields =
co(H,,)
=
inf
CD\H,,)
.
(**)
Quantum Spin Systems
(*)
Hence, by combination of
co(Äa,(A))
-
o,(H^] w'
EXAMPLE 6.2.56.
6
-
CD(H^
{CD'(H^]
inf
-
(**)
and
-
-
one
obtains
^a,(A))
O}'(H^
-
^a>(A))}
C'f
-
inf a/
Let
345
CD'(H^(A))
'
eC^
consider the application of Theorem 6.2.52 to the onenearest-neighbor ferromagnetic Ising interaction, i.e., an interaction which energetically favours the alignment of spins. Since the interaction is classical, it can be argued that a state is a ground state if and only if its restriction to the classical subalgebra is a ground state in the classical sense (see Notes and Remarks). It is also evident that these states are characterized by verifying condition (1) of Theorem 6.2.52 for intervals A [a,b] c Z. But since the only interaction is between nearest neighbors the values of co'(Äp([a,^])) for a given state CD' on the classical algebra are governed by the orientations of the spins at the sites {a l a, a + l Z? l Z?, Z? + 1}, i.e., the only spins external to [a, b] which enter into the minimization problem are those at the neighboring sites a l and b + 1. Now there are two cases to consider, zero external magnetic field and nonzero magnetic field. First, assume there is a nonzero external field, then the energy contribution of the field is minimized by aligning the spins with the field. Thus, for b a sufficiently large, co'(Jff(^([a,b])) is minimized by aligning all the spins, at the sites Z? {, fl + l, l, Z?}, with the field. Consequently, there is a unique ground state. This state has all the spins parallel to the external field and consequently is translationally invariant. Second, consider the case of zero magnetic field. Now the effects of the external Spins a l and b + l are of greater importance in the minimization problem. There are two basic possibilities: us
dimensional model with
=
,
,
.
.
.
,
,
-
-
.
.
.
,
-
(1) (2)
the
Spins at a l and Z? + l are parallel and o}'(ff^([a^ Z?])) is minimized by arranging all the spins in [a 1,Z7 + 1] to have the same orientation; the spins ai a- l and b + l have opposite orientation and o}'(H^([a, b])) is minimized by arranging all spins in [a l jc] and all spins in [jc + l Z? + 1] to be parallel for any x E [a, Z?], i.e., only the pair of spins at the neighboring sites X and ;c + l are anti-parallel. -
,
Thus the external
(1)
states
on
One class consists of two
parallel, (2)
ground
either
positively
or
the classical
,
algebra
now
translationally invariant negatively oriented.
fall into two classes: states with all
spins
The second class consists of
an infinite number of states in which the translation symmetry is broken. Each state has the spins to the left of a given site with one orientation and the spins to the right with the opposite
orientation. The
translationally invariant states describe the two possible modes of spontamagnetization. The nontranslationally invariant states illustrate a form of phase Separation; one side of the System has a positive spontaneous magnetization whilst the second has a negative magnetization. Some properties of these states were discussed at the beginning of the section. neous
346
Models of Quantum Statistical Mechanics
The
model without external field is
Ising
model discussed in
Hamiltonian is
H^(l-n,n])
-jl
=
nature
Ising
model
=
smaller than for the
special
a
|y|
=
l and /l
=
of the X-Y
^1 A
+ 21 X
to
case
Recall that the local X-Y
n
corresponds
of the Jf- 7 model
to
2 ((l +7X0-;+' +(1 -y)^X+')
\^x and the
equivalent
6.2.14A and 6.2.14B.
Examples given by
0. Due to the
n
j
genuine quantum
would expect the multiplicity of the ground states to be model and this is indeed the case. The number of irre-
one
Ising representations, up to unitary equivalence, is äs follows. (a) l if |/l| > l, orif7 0,|/l| < 1. In this case there is a unique ground state which is then automatically translation invariant. If m > l the external field is dominant and it forces the spins to align 0 rotation around the third axis is a symmetry for the along the third axis. If 7 System and this symmetry is not broken by the ground state.
ducible
ground
state
=
=
2
(ß)
In this
longer
if|A|
the external field is weak and rotation around the third axis is
symmetry. The refiection (r]
no
-ff'^.o^]
-^
-crj,
tr^
-^
-^
=
This is the
Ising model without external field discussed above. Two of the four representations correspond to the two translationally invariant ground states in class (1) above whilst the other two correspond to the two classes of non-translationally invariant states in class (2). Each of the extremal ground states in each of the two latter classes can be obtained from one of the others by translation composed with certain inner automorphisms defined by a local unitary. Since the state and the transformed state only diflfer at a finite number of sites the associated representations are unitarily equivalent. If cp is an extremal ground state for the Jf- 7 model, define locally perturbed states by
(pß(A) for all A,B
^^^
with
(p(B*B)
>
=
(p(B'^AB)/cp(B^B)
0. Then the state has the return to
equihbrium
property, lim /
for all A,B ^
ground
^
(pB('^t(^))
^
^(^)
00
Ul^,
state is
if and only if the parameters (y, 1) are unique, i.e., if and only if (a) is satisfied.
in the
region
in which the
Recall that all these one-dimensional models with short ränge interaction have a state for finite ß by Theorem 6.2.47. Using the twisted asymptotic
unique (T, j5)-KMS
abelianness property described in Example 6.2.14B, which is valid in cases (a) and (ß), together with extremality one can establish that all these finite temperature states have the property of return to
We conclude with
ground
a
equilibrium.
discussion of Z^'-invariant interactions and /^'-invariant
states. Our aim is to characterize the latter
by
a
minimum energy per
347
Quantum Spin Systems unit volume
criterion, i.e.,
analogue of
to derive an
the
equivalence (1)
<^
(3)
in Theorem 6.2.42.
First, recall that the mean energy H^ of each Z^-invariant interaction O was äs an affine, weak*-continuous functional over the Z^'-invariant states in 6.2.39, under the mild assumption that Proposition "!', defined
m
=
^^^<^<^^
A'90
One has
//.(co) and hence
|//(i)(co)|
<
||O||
for all
of Z^-invariant interactions functional ^e^^
//((D)
co(g'W) W ;.30
^
=
G
co
E^
equipped
G
inf
-
that O G
inf
//((D)= O)
e
H->
//$(co)
.
//(O)
is
But
concave.
{//xp(a;)+//a)-^(co)}
E^J
inf
<
again denote the Danach space foregoing norm and define the
4^
e
immediately
Let 33
[-||O||, ||O||] by
//((D) It follows
.
with the
{H^(co)^ ||(D-'||}-//(^)
||)-^||
+
.
coeE^ Hence, by interchange of O and ^
one
obtains
\H(^)-H(^}\ <||0-T||, and in
particular H is continuous. The energy function H is the zero-temperature analogue of the thermodynamic free energy discussed in Theorem 6.2.40. This is made more precise by the following proposition which is not essential for the sequel. Proposition
6.2.57.
actions with
norm
Let 23 denote the Banach space
11^11
of
^m^)\\ =
E^7^<+-1^1
;.BO
Let O
H->
F(^)
denote the
thermodynamic free lim
F(
and O
1-^
H(^)
the
mean
--->
energy
|A"Mog l Tr^Je-^^^^))
oo
energy
-^^--kj;:^ ^
It follows that
X3Q
f -invariant inter
348
Models of
Quantum Statistical Mechanics
//(O))
(-ß-^F(ß^))
lim
=
ß
^
OG
all ^ eB.
for
PROOF.
From Theorem 6.2.40
F(ß^)=
{S(co)
sup co
e
ßH^(co)}
-
.
4"
Now for each
ß > 0, choose an coß such that the supremum is attained. This is possible because iS" ßH(^ is upper semi-continuous. Now choose a sequence jß,, such that -ß~^F(ßj^^) converges and a subsequence n' such that coß^, converges in the weak* topology to a state co. By continuity of //$ one has -
lim n'
H^(coß ,}
=
"
>
H(^(co}
.
oo
But
-ß-,^F(ß,^)~H,,(wß^,)\
=
ß',^S(wß^,)
ß~,^\og(N+l)
<
,
and hence
-ß-^F(ß)
lim >
But
äs
the
mean
=
H^(w)
.
00
entropy S is positive
-ß-^F(ß^)
for all co' e
.
This
implies
lim
that
-ß~^F(ß^^)
inf
=
tj-^cG
H<^(co)
=
H(^)
.
we^f
Thus, all limit points of -ß~^F(ß^) exists and is equal to H(^).
are
equal
to
H(^)
äs
ß
^
OG.
Hence the limit
The zero-temperature analogue of Theorem 6.2.42 is derived by the same arguments used to prove (3) => (1) in this theorem coupled with the previous
stability result, Theorem Theorem 6.2.58.
6.2.52.
Let ^ be
m,
a
f -invariant interaction such that
=
Y,mx)\\e^\^\<
+^
X 30
for
/l > 0 and let
some
If 0} (1) (2) PROOF.
is
a
co
is ßT^
co
minimizes
(1)
T^ denote the associated dynamical
f -invariant state the follomng
=>
(2):
groiind
Let
are
group.
equivalent:
state.
H(^.
co
be
a
Z^-invariant T^ ground state. Theorem 6.2.52 implies
that
CO(^CD(A))
<
co'(Äo(A))
349
Quantum Spin Systems for all c/y e
Now let
O'^.
a
G
E^^
and define co'
oJ
=
(j|,i^
co(Äi>(A)) But
//CD (A)
äs
=
//(I) (A) +
oX//(i,(A))
and
?f"ci){A) <
0
oj
by
.1^^.
.
oy(Äi>(A))
<
fT(//ci)(A))
-:
.
oy(//ci)(A)),
ö-(//ci,(A)) +oy(^ci>(A))
-
this
gives
c/j(J'Fci)(A))
(A)||. observed in the
of Theorem 6.2.42 that
Now
we
already
A
->
oo.
Hence, dividing both sides of the last relation by
A
>
oo
proof
|A|
||?'f''(D(A)||/|A| and
taking
^
0
äs
the limit
yields
//ci)(oj)
<
//O(CT)
.
Thus
//ci,(cü)
inf
-
a
(2)
=^
(1):
The
e
//ci)((j)=//(0)
proof of this implication (1) in Theorem 6.2.42.
used to prove (3) =^ by the mean energy
.
,; is
a
One
repetition of the convexity argument simply replaces the free energy F(^)
-//(>).
The
foregoing theorems give general characterizations of ground states and ground states but in particular models one can give a much more detailed and precise description. In Example 6.2.56 we described all ground states of the one-dimensional ferromagnetic Ising and Jf-F models and we end this discussion of spin Systems with some comments on the translationally invariant ground states of the one-dimensional ferromagnetic Heisenberg model. If there is a nonzero external magnetic field, it follows from Theorem 6.2.52, by the argument used for the Ising model in Example 6.2.56, that there is a unique ground state and this state is translationally invariant. If, on the invariant
other
liand, there is
external
field, then there exists a continuum of invariant is a product state in which all the spins are but there is a continuum of possible orientations. Let us briefly describe one of these states and the corresponding dynamics. We consider the state co determined by the configuration in which all spins are down and we begin by describing the associated representation (^^j, TÜ^J, ^oj) but, for notational simplicity, we drop the symbol TT^,;, i.e., we identify n(,j(A} and A. Since all spins are down the cyclic vector Q^j is a simultaneous eigenfunction of all the cr^, .x G Z, with eigenvalue 1. Next reintroduce the spin raising and lowering operators cf^ (er] (T\)/2 of Example 6.2.13. One has no
states. Bach of these states
ground parallel
=
[4,<^3]-T24, and the
readily and
a
corresponding
that
(f^_^,o
=
K,
to different x, j G Z
0, for all
z
G
Z, and the
=
a-^
mutually
vectors
\X}
commute. It follows
deüned
by \0}
=
ü,
350
Models of
Quantum Statistical Mechanics
n<
W-
\x^X
form
orthonormal basis of
an
^^
Theorem 6.2.48. Thus if
{\X)]X
Z,\X\ =n}
C
one
^.
This basis
denotes the
was
used earlier in the
subspace spanned by
proof of
the vectors
has
.
© öL'^
-
n>0
The vector
Q^o describes the configuration with all spins down and the vectors correspond to configurations in which exactly n spins are up. Note that the representation (^, TT^J, Q^) is analogous to the Fock representation of the CCR and CAR algebras described in Chapter 5 with the spin raising and lowering operators repladng the particle creation and annihilation operators, the n-spin subspaces ^^ replacing the n-particle subspaces, and the raised spins analogous to the particles. Analysis of the Heisenberg dynamics exploits this analogy and interprets the raised spins äs some form of quasi-particle. These quasi-particles are usually referred to äs spin waves or spin excitations, The dynamical group T corresponding to the Heisenberg dynamics is represented on ^ by in
^^^
A^it(A]=e^'^-Ae-^^^where the
selfadjoint
Hamiltonian /4j is defined
// of bounded
selfadjoint
=
Hamiltonians
Hl^^\X}
=
,
äs
the direct
sum
//w H^'
whose action
on
9)^^
is
given by
-jY{a^-
infinite sum causes no difficulty since the a at different points commute 0. Moreover, //i"^ > 0 for / > 0 because ^ (f a^+^ (ö^ (T^+i 1])|0) and 0 The action is o^H^^ One has eigenvalues 4.) particularly simple.
(The and has
.
-
-
-
-
/4^)|x)^[/4^<]|0) --2J+^-2(r;
+
--2J(|x+l)-2|x) i.e.,
H^^
(r;-^)|0) +
acts üke
|x-l))
,
a multiple of a double difference operator. Hence, the evospin vectors, i.e., the elements of ^^\ is a lattice analogue of the free evolution of point particles discussed on Fock space in Chapter 5. In particular if / G ^^(Z) and
lution of the
one
^+(/)=
E/w<' ,TeZ
one
calculates
by
Fourier transformation that
Quantum Spin Systems
e''^"'
=
X" a+(/)|0)
351
cT+(/,)|0)
=
where
f,(x)=
Y,G,(x-y)f(y] yel
and /ITT
Gt(x]=^r dOe^^^^^-^'^^^ Thus the
one-spin evolution is a free motion with lattice momentum ö G [0, 2n] corresponding energy 4J(1 cosö). Comparison with Example 6.2.14 shows that this is essentially the evolution of the Ä'-Fmodel with an additional energy 4y contributed by the Ising interaction. The evolution of the multispin States is more complex but can be thought of äs a free evolution stemming from the X-Y contribution to the Hamiltonian together with a spin interaction arising from the Ising contribution. Adopting this point of view one defines a free multispin evolution ? G IR f-^ t/^P by setting and
a
-
f/
.
.
.
C7+(/W)|0)
=
a+(/(")
.
.
.
,
is the one-spin evolution introduced above and Uf is extended t \-^ f^ by linearity. One then exploits scattering theory methods to obtain properties of the Heisenberg evolution Ut Qxp{itH(,o} by reference to the free evolution Uf. Both groups describe the total evolution of complexes of spin excitations together with the relative motion of these excitations. The Ising contribution to
where
//co appears leads to
äs an
interaction between the various excitations. This interaction
variety of phenomena, binding of two or more spin excitations, scattering of bound or unbound complexes, etc. We will not enter into more details of the mathematical description of these problems but remark that the physical picture of spin excitations, or spin waves, has been fruitful both conceptually and analytically. For example, the realization that H(^ describes an Overall free motion of a System of interacting spin waves indicates that H(o has absolutely continuous spectrum, on the orthogonal complement of O^;. This has indeed been verified (see Notes and Remarks) and consequently a
lim \t\-^oo for all (p,
i^
G
(iA,e^'''>)-((p,Qc.)(ßa.,(p)
öc^. Alternatively stated, lim \t\-^oo
o}(AT:t(B])
for all A and B in the
spin algebra, i.e.,
the
4.
sense
of
Chapter
=
the
(D(A)o}(B)
ground
state
CD
is
strongly mixing
in
Despite a great deal of analysis of the translationally invariant ground states Heisenberg model, little attention has apparently been paid to the examination of nontranslationally invariant states and even a proof of existence of the
352
or
Models of
Quantum Statistical Mechanics
absence of such states appears to be lacking. If one considers the state co by the configuration in which all spins to the left of 0 G Z are down
determined and the 0
spin
and all
spins to the right of 0 are up, then this state is not a ground Heisenberg dynamics but is a ground state for the dynamics obtained by removing the interaction between the Oth spin and the Ist spin. Since this interaction is given by a bounded element of the spin algebra, one can use the perturbation techniques developed in Chapter 5, e.g., Corollary 5.4.2, to deduce that the füll Heisenberg dynamics is covariantly implemented in the representation (,^,7üaj,Ocü), i.e., there is a selfadjoint operator HO, such state of the
that
7i,,(i,(A}} where
e^^^-n,,(Ä)e-^^^-
,
denotes the
T
perturbation tems
=
one
Heisenberg group. Since H^o is constructed äs a bounded of the positive ground-state Hamiltonian of the decoupled Sys
deduces that Hf^ is lower semi-bounded. In fact H^o may be explicitly The state co has the form ^^EZ^PX where cp^ (p__ for ;c < 0 and
computed. (Px
=
9+ for
tensor
=
X
>Q, The
corresponding representation
space is the
incomplete
product
r^-^.xez(C-,Q,0 NO W the nearest
neighbor operator
D
-
(f
-
.
(f^^ has eigenvalues 0 and 4
on
C" (g) C" and the operator is in fact 4P_ where P_ is the orthogonal projection onto the antisymmetric subspace of C^ 0 C^. Thus the formal Hamiltonian,
i.e., the
sum
of the translates of the l
-
(f^.cf''^^,
from the reference vector <^x^x at the reference vector itself, except at the kink
differing space is
a core
not have
expressed
jc
=
0, -l,
one
in this form is
obtains
zero.)
This
But H^o does
positive. eigenvalue because the corresponding vector would have to single term of the interaction and thus be permutation invariant.
zero äs an
minimize each But
for H.o^ Thus H^j
converges on the set of vectors only a finite number of sites. (On
is the infimum of the spectrum of //^j. To deduce this consider product coinciding with Q_ on the left of a finite set of 7V consecutive points, with O+ to the right of the set and rotating through an angle n/N between neighboring points of the set. The contribution to the energy of two neighboring points is proportional to sin^(7i;/7V). Hence the total energy is propor zero
vectors
tional to
Nsirr(7i/N}
which tends to
zero äs
A^
^
oo.
This argument indicates ground states for the
that there may not be any non-translationally invariant one-dimensional Heisenberg ferromagnet.
6.3. Continuous
Quantum Systems.
II
Quantum spin Systems give a relatively simple explanation of the most important magnetic phenomena. In Section 6.2.2 we demonstrated that the Ising model exhibits
ferromagnetic behavior if the interparticle forces favor spin alignment and antiferromagnetic behavior if it is energetically more favorable to have anti-parallel neighboring spins. If one reinterprets this model in the lattice gas language described at the beginning of Section 6.2.1, one obtains a rudimentary description of liquid-gas and solid-liquid phase transitions. In this language a positively oriented spin corresponds to a lattice point occupied by a particle and a negatively oriented spin corresponds to an unoccupied site. Thus, the ferromagnetic phenomenon, the favoring of large blocks of parallel spins, can be interpreted äs the formation of large blocks of occupied sites, "liquid droplets," interspersed between large blocks of empty sites, "gaseous regions." On the other hand, anti-ferromagnetism corresponds to the forma tion of a periodic structure in which one sublattice is occupied by particles and the other sublattice is empty. Thus, one obtains a primitive version of crystalline structure. Note that the ferromagnetic case corresponds to an attractive interaction between neighboring particles while the anti-ferromagnetic case corresponds to repulsive forces. It is widely beheved that a more reaHstic and detailed description of the solid-liquid-gas transitions should be obtainable from analysis of continuous models with interparticle interactions which are partly attractive and partly repulsive. Unfortunately, there are few rigorous results to substantiate this belief. Typical interatomic, or intermolecular, forces are attractive at large distances and repulsive at short distances. Thus, it is plausible to suppose that at low densities, for which the interparticle spacing is on average large, the attractive forces would give rise to the formation of hquid droplets in the manner observed for the lattice gas. Alternatively, at high densities for which the interparticle spacing is small one could well expect the interparticle repulsion to force the formation of a periodic, or crystalline, structure. Un fortunately, one is far from a quantitative justification of these simple quahtative ideas. In fact, the only transition phenomenon which has been rigorously derived for nonrelativistic continuous particle Systems of the type we consider is the Bose-Einstein condensation of the noninteracting Böse gas. This was described in detail in Chapter 5, Section 5. It is an effect of quantum statistics and not of particle interaction. It is believed that this type of con-
354
Models of
Quantum Statistical Mechanics
densation will still
prevail in interacting Systems and that it is directly related to important physical effect, superfluidity. Unfortunately even the existence of Bose-Einstein condensation in interacting Systems has resisted proof. Thus, the rigorous theory of equilibrium states of continuous Systems is much less complete than the corresponding theory for spin Systems. Nevertheless, some results are known about existence of equilibrium states at high temperatures and low densities. The purpose of this section is to partially review another
these results and
some
of the
new
methods which have been used to obtain
them. In Section 5.2 we studied the equilibrium states of quantum Systems of noninteracting point particles in R^. The description of the Fermi gas was particularly simple because the time development is given by a strongly con tinuous one-parameter group of *-automorphisms T of the CAR algebra and the Gibbs equilibrium states coincide with the (T,^)-KMS states where ß G [Ru{it 00} is the inverse temperature. These states are readily computable, see Examples 5.3.2 and 5.3.20, and one finds that there is a unique equilibrium State for each ß. The Situation for the noninteracting Böse gas was much more complex because the time development is given by a discontinuous group of *-automorphisms T of the CCR algebra. To make global Statements about the set of equilibrium states (see Example 5.3.2), it was necessary to impose some extra continuity conditions. Within this framework, however, the equilibrium states once more correspond to the (T, jß)-KMS states. Moreover, no KMS State occurs for ß < 0, there is a unique KMS state for ß 0, and nonuniqueness arises if ß > Q. This nonuniqueness corresponds to the Bose-Ein =
stein condensation.
The Situation for interacting quantum Systems is certainly much more complex. First, no global Statement of the time development äs a group of *-automorphisms of an appropriate C*-algebra has been obtained and, second, there are plausible physical reasons for believing that such a development, if it existed, would be discontinuous in the norm topology. The problem is that a norm-continuous time development would imply that every state of the System evolves continuously throughout time. Nevertheless, one can easily envisage states for which discontinuous phenomena occur after a finite time, e.g., states for which an arbitrarily large local density or an arbitrarily large local accumulation of energy occurs. Thus, in this respect the noninteracting Fermi gas appears to be atypical. These problems with the time development can be partially circumvented by using the thermodynamic limit procedures employed in Theorems 5.2.24 and 5.2.32 and the associated remarks. The idea is to examine the limits of the
finite- volume Gibbs states and the associated tions. If
60 A
time-dependent Green's funcnoninteracting gas in the bounded set local time development, and T the corre
denotes the Gibbs state of
A, 0) an infinite-volume limit, T^ the sponding thermodynamic limit, then
co(ÄQ^t,(A,)
-ijA))
=
a
we
established that the Green's functions
lim A
>
00
COA(^O<(^I)
-^f (A)) "
Continuous
exist for all
AQ,.
.
.
particles,
II
355
^^ and t\,. ,tn ^^. Now if one considers interacting the thermodynamic Green's functions examine to possible
,A
then it is
Quantum Systems.
.
.
equilibrium phenomena and the time questions in more equilibrium. development detail in Section 6.3.3. Prior to this however we discuss the general definition of the interacting System and its equilibrium states. We also introduce and analyze various functional integration techniques which are of great use for the analysis and to obtain
some
Information about
of states
We will discuss these
near
of these states.
6.3.1. The Local Hamiltonians We first discuss the existence of
and the Gibbs
dynamics
equilibrium
states for
bounded open subset A of the configuration space R^. Since the discussions of the Fermi gas and the Böse gas are very similar, we treat the two cases simultaneously. For simplicity we
Systems of interacting particles confined
to a
spinless particles although there is no difficulty in principle in considering several species of particle or particles with spin but with spinindependent interactions. Let L^(^Y denote the subspaces of Z^(A") formed by the totally Symmetrie (plus sign) and totally anti-symmetric (minus sign) functions of n variables X G A. These Hubert spaces describe the quantum states of n point particles obeying Böse statistics (plus sign) or Fermi statistics (minus sign). The assoconsider identical
ciated Fock spaces
5(A)
=
0l2(A)'^ n>Q
an arbitrary number of particles (for more details see particular Example 5.2.1). Next let ^(A) denote the CCR algebra (plus sign) and the CAR algebra (minus sign) over the one-particle space Z/^(A). The action of these algebras on 5_^(A) is described and discussed
describe the states of Section 5.2.1 and in
in Section 5.2.
The time evolution T^ of a System of noninteracting particles is a oneParameter group of *-automorphisms of ^(A) implemented by a unitary group
t^Qxp{itdr(TJ^^)}, TJ^^
selfadjoint extension of the Laplacian -V^ on CQ(A). One can variety of T)^^ corresponding to different boundary conditions, i.e., corresponding to different dynamical behavior of the particles at the boundary 9A of A, but for simplicity we restrict our attention to the classical boundary conditions described in Example 5.2.26. In particular we now use T^ l to denote the selfadjoint extension of V- corresponding to the conditions d\l//dn ffil/ where a G C^(9A) and we assume that the boundary 9A of A is piecewise differentiable. Moreover, we also assume that er is non-negative and
where
consider
=
a
is
a
wide
356
this
Models of Quantum Statistical Mechanics
T^ ^
that
ensures
quadratic
>
0. This last result is
r^^|
form characterization of
given
an
easy consequence of the
in
Example 5.2.26, i.e., for
Aez)(r(;))
l dx\V^(x)\^+ l
(^,T(>)=
JA
>0
Finally,
we
T^^^
let
the Dirichlet
.
denote the
boundary
selfadjoint extension of V^ corresponding to 0 on 9A. For this latter extension it is i/^
condition
=
not necessary to assume smoothness of
Next
dxa(x)\^l>(x)\^
JöA
9A.
define, T^^A by
r,,A
=
jr(r^;^)
and let
T-^A
T-S
=
>0
be the
decomposition of T^^A corresponding to the direct sum decomposition g^(A) into the 77-particle subspaces L^(A)^, i.e., T^^]^ is the restriction of T^^/^ l2(A)^. Alternatively stated, T^^^^ is the restriction to I"(A)^ of the closure
of to
of
the operator.
(rf^^i 0 n (g) acting
on
with the
n)
(g)
L^(A.Y. Thus,
positive
closed
+
(D
(g)
r^l^ (g)
(8)
-
n)
+
(n
+
(g) n 0
0
r^|j)
r["| is the non-negative selfadjoint operator associated quadratic form defined by .()
-
~
V,A
/-)
^0,A
l
~^
o(^) ^(7,A
'
where
fi('/')
=
[
-'A"
dx,...dxY^ ,=
4"i('/')
=
^ [dx,... ^=1
and
f
JA
|(V,,(xi,
.
.
.
,x)p
,
1
dXi...
JöA
[dx
(T(^,)I'A(^1>
.
.
-
,^)P
,
JA
D(r"\} consists of the (l/ G ^^(A) ^ for which each derivative lies in Z^(A)^. ^^"]^ is positive form is evident from the assumption > 0 but it
The fact that
requires
a
er
work to prove that it is closed. In the Standard result of partial differential equations that we
Example
some
5.2.26. The argument for
general
n
l, this is
case
already
a
mentioned in
is very similar and
we
omit the
details.
Next, erator
we
wish to introduce
corresponding
to the
particle interactions by the addition of an op potential energy. Thus we try to define a selfadjoint
Hamiltonian. ff
ff, A
=
^ff,A
+
^A
Continuous
and then the evolution of
an
=
is defined
M(A)
observable A E
Tf(^)
Quantum Systems.
e^^ff^A^e~^tffa,^
II
357
by
_
only consider interactions coming from potentials and hence the number of particles is conserved. Thus, the total Hamiltonian has a decom-
We will
position ^^.A
e
=
>o
in terms of
-particle
Hamiltonians
//^"|
L^(A)^.
on
These latter operators
have the form rr() _
^t7,A where the interaction operator
-
UJ^^
j^(n}
is
r
,
^f7,A an
+
An]
^A
'
operator of multiplication,
(t/i"V)(^i,...,^)-^^"H^i,...,^)^(^b..-,^) by
real function whose values
a
U^"\xi,.
.
.
,Xn) correspond
to the
potential
,jc. Note that for t/^"^ to preserve energy of n particles at the points jci,jc2, it is the symmetry, or anti-symmetry, properties of the elements in .
.
.
L^(^Y^
necessary that it is
a
totally Symmetrie
function
i.e.,
^(")(xi,...,x,)-t/(")(^,(i),...,x,(,)) for all jc/ e K" and all permutations TT of {l, 2, , TI}. In Order to be more precise in this formulation one must consider several .
.
.
different details. It is necessary that //^^A is selfadjoint and for this one must in such a + specify the interaction U^"^ and the meaning of the sum
T^"\
way that
direct
//;"|
sum
of
U^^^
selfadjoint. One must also clarify the meaning of the infinite the selfadjoint //^"l- This latter point is, however, straightforis
ward.
{//^"^}^j> j be an infinite sequence of selfadjoint operators with domains {)(//("))}>! in the Hubert spaces {^"^}>i. We define the infinite direct sum Let
H^@ //() n>l
by
/)(//)
=
1.A
=
{^()} >,;,/.() D(//W), 53
l
>i
||//("V'">||'<+oo|J
and
//lA
=
V*"' l {//(" l J
for
il/
=
{jA^"^}>
^W- It follows that H is selfadjoint. To deduce this, first obviously a densely defined Symmetrie operator. But the closed by the Standard argument that proves that the infinite l
remark that H is
graph
of H is
n>l
^
358
Models of Quantum Statistical Mechanics
direct of 11] ;?,
sum -
of Hubert spaces is complete. Thus, H is closed. Finally, the ränge for each A G C\R because R(n equal to //('^)) ^"^ for each
// is
=
-
i.e.,
R(1^-H)= @R(^^-H^''^)
0^"^-
=
n>\
//
>
.
l
This establishes the
selfadjointness of the direct sum. Hence, the initial problem of existence of dynamics is reduced to the discussion of selfadjointness of the n-
particle Hamiltonians ^^''|. There is a wide variety of methods for defining Hamiltonians of the type under consideration and studying their selfadjointness, e.g., quadratic-form techniques, pointwise positivity arguments, perturbation theory, and functional integration. Each method has its own peculiar advantages and the appropriateness of any given technique is governed by the properties of the interaction in question and the specific problem under consideration. In the remainder of this section
we discuss various aspects of the selfadjointness problem for the //^'^| by use of form techniques and perturbation theory. In the subsequent description of equihbrium states, it is however more appropriate to define the Hamiltonian by an implicit functional integration technique. Under
suitable restrictions
Hamiltonian and
on
the interaction these methods all lead to the same then pass from one method to the other. This will be
one can
discussed in the next section but first it is necessary to be cerning the interaction. The most
lationally >
over
more
specific
con-
commonly studied interactions are those that arise from transtwo-body potentials. A two body potentiell i^ a real function
invariant
R^'
x
[R^ whose values
Thus, the total
represent the potential energy of in point x\ and a second particle at the point X2. interaction energy of n particles at the points xi,.T2, ,;c is
teraction between
a
particle
at the
.
.
.
given by
f/(")(xi,...,^,)-
E l
Note that the symmetry of
U^^^^
^(^-^y).
is reflected
by
the symmetry property
^(Xi.Xj) =^(xj,Xi) of ). The
Potential is said to be translationally invariant if (x/, Xj] is a function interparticle position xt -xj. Thus a translationally invariant two-body Potential is a real Symmetrie function O over U^' and the associated interaction energy U^"'^ is given by of the
t/W(x,,...,X)=
^
^(Xi-Xj).
\
The total interaction energy is then
U^=^ JA
dx^
/ JA
formally given by
dx2^(xi -X2}a*(xi}a''(x2)a(x2)a(xi)
,
Continuous
Quantum Systems.
359
II
Example 5.2.1. Similarly, one can define a (translationally invariant) k-body potential äs a real-valued totally Symmetrie function O^^^ over (R^)^ (invariant under simultaneous translations of all k variables in R^') and then
where
we
use
the notation of
rr()/,., \X\^
U
\
.
.
., ^Xn)
.
_
The sequence O
=
Y^ /
>P /
{^^^^}^^> i
=
l
(^(^^(x^ V-^/n
^
^
A:
X- ) 5-^/A-;
l
is the
of the interactions introduced for
analogue
quantum spin Systems. Note that there is
a converse
to the above construction, If
one
U^"'^
is
given
a
(U^')" totally Symmetrie {^^"^},j>i sequence U representing the interaction energies, then one may introduce Ä:-body potentials by recursion, e.g., functions
of real
=
^(^\x)
=
0(2) (;Ci, X2)
=
U^^\x}
over
,
t/(')(%l, X2}
-
U^^\X,)
-
U^^\X2)
,
etc.
A
typical translationally invariant two-body potential which physics is the Lennard-Jones potential
occurs
in
atomic and molecular
^(^^=Rf^-7^' 0,6 > 0. The positive term a/\x\^^ predominates when two atoms are dose, i.e., when \x\ is small, and this corresponds to a repulsive force arising
where
negative term b/\x\^ pre large interatomic spacing and this corresponds to an attractive force originating in the electrostatic interaction of the charged subatomic particles which build the atom. This attraction at large distances is called the van der Waal's force. The principle features of this potential which we abstract in the following are that it is lower semi-bounded and locally integrable away from the origin. The lower-semi-boundedness property is of importance because it allows the application of quadratic form techniques in the definition of the local Hamiltonians //^"|. There are however important physical potentials which do not have this property. The Coulomb potential between two particles with charges Ci, Cj, at the points jc/, Xj, G (R^ is given by from the interaction of the atomic nuclei. The dominates at
%(x,--x,)=r-^ |A/
and this
can
be
arbitrarily negative
Nevertheless, effects of this kind
can
if the
charges are of the opposite sign. by perturbation techniques.
be handled
First, however, let us consider the lower-semi-bounded case. Throughout the sequel it is of importance to recall that there is a one-to-one correspondence between positive selfadjoint operators T on the Hubert space
360
Models of Quantum Statistical Mechanics
and
is
positive, densely defined, closed, quadratic given by
^(^) the
H this
identity
tween
the
correspondence
correspondence
||r'/2^f
=
D(t) =Z)(r^/^). Furthermore, by
with
forms t. This
the addition of
multiple of correspondence be-
constant
a
may be extended to
a
lower-semi-bounded T, and lower-semi-bounded t, i.e., if T
correspondence
is
t(^]
=/)((r-hc1])^/")
withZ)(/)
We define
>
-cH, then
given by =
\\(T+c^Y'^m^-cm'
and
one
has t>
-c.
lower-semi-bounded interaction energy to be a real-valued totally U^""^ of n variables in (R^ satisfying the two conditions. function Symmetrie a
f/("H^b ^^n) >-Cn for some c^ > 0 and all xi, ,x G R\ t/(") G L\^^(W\Sn) where iS is a closed set of Lebesgue measure
(1)
.
(2)
If the constant c can be chosen to be zero, Note that the second condition means
we
call
U^''^
a
.
.
positive
zero.
interaction
/J^i...^^|t/(^)(xi,...,z,)|<+^ JK
for each compact subset K c the interaction potential. For Lennard
-
Jones
S^ The
requirement
with hard
the
cores.
potential
{(jci
,
.
.
.
IR^^ViS.
The set Sn allows
possible singularities in arising from the
the interaction energy satisfies these conditions with
,x);
example,
jc/ G
[R^ and jc/
that Sn should have
=
Lebesgue
As the inclusion of hard
cores
jc/ for
some
/
^ j}
measure zero
.
excludes
particles
entails additional discussion of
boundary conditions imposed
upon the cores we have made the restriction irrelevant details. In fact, the discussion of hard-core particles is easier in some respects than the discussion of point particles because the hard core prevents a large number of particles collapsing into a small volume. on
Sn
to avoid
Next
we
^"^ and associate to each lower-semi-bounded interaction energy U a selfadjoint interaction operator by use of the quadratic-
L^^"^
each A c R^
form
techniques described
above and in Section 5.2.3. First, define the
quad
ratic form
U^^\^}=
l
dx,,..dXnU^''\X,,,..,Xn)\^l^(x,,...,Xn)\^
JA"
where the domain
integral
D(u\^)
of the form consists of all
is finite. This form is lower
it is also
densely
defined since
compact Support in
semi-bounded,
D(u^^^)
ij/
w^^
G
^"(A)^
><:,
for which the
and closed. But
contains all bounded functions with
([R^"\5')nA". Thus,
there exists
a
lower-semi-bounded
f/|"^ > -cj such thatD(w^"^) Z)((t/^"^ +cj)^/^) 4^^W-ll(< + -J)^/Vf--
selfadjoint operator
=
and
Continuous
U^^^^
We refer to
by t/^"^
äs
361
.
forms.
note that the sum of two lower-semi-bounded closed
First,
II
the interaction operator It is the operator of multiplication L^(A)^. Now we can define a total Hamiltonian by summing
on
quadratic is
Quantum Systems.
quadratic
forms
automatically lower semi-bounded and closed. Thus, if v^\ denotes the po quadratic form associated with the noninteracting Hamiltonian
sitive closed
T^^\,
then the
sum
^(n] _^(n] '^a,^
,
v,A
^
(n] "A
densely defined
is lower semi-bounded and closed. But it is also
domain contains the
Support in
operator
once
h"^\
([R^"\iS')nA'^.
selfadjoint operator
H^^\
W^\
in the canonical fashion. This Hamiltonian
c1l
>
is called the form
sum
L/W
^(T,A The form
sum
it could well sum
differs in
occur
would have
that
because its
continuously differentiable functions with compact also determines a lower-semi-bounded Thus,
_
of
T(")
^ff,A
T^"^ _L +
and
[/^"^
and
we
write
r/^") ^A
from the operator sum. In the case at hand is empty and hence the operator n in domain of definition. To ensure that its vector
general
D(T^"l)
no nonzero
Z)(t/^"^)
the operator sum is densely defined it is necessary to place more stringent restrictions on U^^\ Then it is often possible to prove equality of the two types
example, if U^"'> e L^(A.Y and f/^") > 0, one can establish that r["| -f U^^^ essentially selfadjoint on C(A'^) n Z)(r^,A) and hence the form sum and the (closure of the) operator sum coincide. (See Notes and Remarks.) In Section 3.1.4 we considered relatively bounded perturbations of generators of semigroups and in the present context we specialize to perturbations of a selfadjoint operator. Let 5 be a selfadjoint operator and P a closed Symmetrie operator on the Hubert space , then P is defined to be relatively bounded with respect to S, or S-bounded, if D(S) c D(P} and
of
sum.
For
is
\\Pi^\\
\l/
e
D(S)
and for
some
a, Z? > 0. The relative bound of P with
respect
to
bound of this type is valid. It is now an easy consequence of Theorem 3.1.32 that 5 + P is selfadjoint whenever the relative bound is less than one, i.e., the above inequality is vaUd with b < l. S is the infimum
Moreover, if S
perturbation
>
over
all b for which
0, then S
+ P
a
is lower semi-bounded. This follows from the
series definition of the resolvent of 5' + P,
(^ + P +
al)-^-(5 + al)-^^(-P(^ + al)-^)" >o
for
a
>
0, and the relative bound which gives
||P(5-f Thus,
one
has convergence of the
a1l)~^||
perturbation
fl/a
.
series for all
sufiiciently large
a.
362
Models of Quantum Statistical Mechanics
Now let
US
functions U^'^^
then if
consider interaction operators Uf^'' of multiplication is relatively bounded with respect to L-(A) ^^, If
t/|"^ \\u^^\^\\ < alliAIK^IlrgiAll,
on
by choosing i/^
to be
constant one
a
deduces that t/^''^
G
Uj('^
by real
TQ^'^V, i.e.,
L-(A)'_]_. Similarly,
is bounded with respect to one of the other classical kinetic energy it is certainly square integrable in compact subsets of the inoperators terior of A". Thus, some form of square integrability is to necessary for the be relatively bounded by the but no sufficient conditions appear to be known. It is possible, for translationally invariant two-body interactions in low
7;"]^,
L^|"^
T^^\,
dimensions, that
square integrability is both necessary and sufficient for lative boundedness with respect to the Neumann operator. This is indicated
the
re
by
following.
Proposition 6.3.1. a real function U If v
< 3
Let on
U/^, where A
L^(A),
C
IR^', be
(Uj\il/)(x}
i.e.,
then the following conditions
=
are
operator
an
of multiplication by
U(x)ili(x). equivalent:
(1)
ueL^(^).
(2)
U/, is
(3)
t/A is TQ l -bounded with relative boiind
T^Q^l-botmded. zero.
PROOF.
Condition (3) obviously implies (2) and this latter implies (1) by ceding discussion. Thus, it remains to prove that (1) => (3). If l// e C^(A), then
the pre-
||t/AiA||<||^||. supl^WI. ,TeA
Next let with
{jA/}/>i denote a complete orthonormal family corresponding eigenvalues {fi/}/>i. One has 'AW
=
E7TioA"(4'i ^.>l/ +
for each
a
>
0.
+
of
i)'A)
a
Therefore,
'^.c^:
hAwP<Eäii(4A+i)'/'iP ;it(e/+r
by
the
Cauchy-Schwarz inequality.
Thus
l|t/AlM|
+
|i<'],AII),
where
c-supfvI'^'-Wl""' ^oi
^up
l
^
^
^A\f^,(8,+a)-
eigenfunctions
of
T^^
Quantum Systems. II
Continuous
This
Q
0
>
äs a
proof
is
closure.
D(TQJ^) by if
complete
one
Thus,
show that
can
CXD.
-^
This last
of
all
to
if Ca < + cxD, and the
D(U^),
^
extends
immediately
inequality
D(TQ^\)
363
is
point
particularly easy for special choices of A, e.g., parallelepipeds, ij/f and e/ can be explicitly computed. To handle the general
spheres,
etc., because the
case we
remark that
Er^=r^^'5^-''>o,A(.,.;, f^(/ a)7o
+
where
denotes the kernel of the
p(^^\(x,y]ß]
known^ that there exist positive 0
for all
A and hence
jc
estimates that c
immediately extend
This result does not
semigroup ß
c^
c'f^
and
i-^
expl-^r^^}.
But it is
such that
<4"r"+tf
po,A(x,x;js)
<
one
constants
^
0
äs a
oo.
^
to all classical
boundary conditions,
e.g., for Dirichlet boundary conditions it is possible for U to have a singularity on the boundary 9A of A. Nevertheless the same argument shows that U G
^^(A)
is sufficient for ^A to be
Now let One
relative bound
zero.
Hamiltonians.
/i-particle general interaction Hamiltonians by combination foregoing techniques. If U^ is a 7"^'^^ relatively bounded interaction
can now
of the two
r[^]^-bounded with
return to the discussion of
US
define
more
with relative bound less than
action defined
one
V^
and if
form methods, then
by
is
can
one
a
lower-semi-bounded inter
introduce
a
selfadjoint
total
Hamiltonian L7('0
^a,A the operator For this it is
by first taking sum
and
with
F|"-
.
D(T^'\ + U^^^)
=
/
7-('0
(VA
_
-
of
sum
, +
T^'^^
important
j(n] ^A ^ \
T
\
+
T
An]
^A
U^^^^ and T^J^\ -\- U^^'
and
that
D(T^'\]. This last property
subsequently
the form
is lower semi-bounded
ensures
that the form
sum
is
densely defined. It is possible to introduce more general Hamiltonians by use of additional techniques such äs relatively bounded forms, etc., but for brevity we restrict ourselves to the above class.
Selfadjointness
of each H /. ^(7,A
ensures
that the total Hamiltonian
//.,A-@^'"^ ^.A "
-
'^
/z>0
is
selfadjoint
observable A G
equilibrium
more
^(A).
state.
trace-class^ for
^For
a
details
It is not however sufficient for
For this it is necessary that
sufficiently large
on
'^Here
set of values of
this and related
and the references cited in the
multiples
development of each defining the local Gibbs
and this is sufficient to define the time
semigroups corresponding Notes
see
we
have reintroduced the number operator N/^
of the identity l
Z.^(A)'_J_
the
sum
j5
>
/IA^A)}
is of
0 and /x G [R. Hence it is
Section 6.3.2, in
particular
Theorem 6.3.8,
and Remarks.
l
on
exp{-j5(//(7,A
Ha,\
on
Fock space. Since A^A is
a
direct
IJ.N\ is automatically selfadjoint.
sum
of
364
at
Models of
Quantum Statistical Mechanics
least necessary that
follow from
i.iN\ is, lower semi-bounded. But this does
//^^A
not
H^'^\
present assumptions on the interaction. Although each is bounded below, the direct sum could fall to be semi-bounded if the bounds c^H on are not suificiently uniform. In a physically stable System one expects the energy per particle to be bounded below since if no such bound existed, the our
H^^\
System would collapse into
a
bound state with
large negative
energy. Thus
one
expects
^i1
-Bn^
>
(7, /
for
^ >
some
0,
or,
equivalently, H,,A
>
-BN^
.
Since the kinetic energy T^^A is positive, this stability condition would be sured if the interaction satisfied the conditions
en-
U^"\x,,...,xn}>-Bn for all
77
=
1,2,
.
.
.
,
and all x/
U\ Interactions with this property
E
are
called
stable interactions. The simplest type of stable interactions arise from positive potentials but there also many interactions with partially negative potentials which are nevertheless stable.
EXAMPLE 6.3.2. variant
Assume the interaction [/^"^ two-body interaction potential >, i,e.,
[/(")(X1,...,^)^
^ l
is defined
by
a
translationally
in
^(Xi-Xj],
and that > is bounded below.
and
Furthermore, assume there exist numbers 0 < öi < 2 positive decreasing functions i and O2, on (0,ai) and (fl2,oo), respectively,
such that /'öl
/ 70
rOO
dt^^(ty-^
=
/
+00,
dt^2(tY-^
<
+00
.
Ja2
If
(D(.T)
>
a)](W)
forW
and
)(.T) then
t/^"^ is stable.
that if A is
a
In
cube of
fact,
-^2(\A}
>
more
foi-|x|
> 2
,
is true. The interaction is
sufficiently large
volume
|A|
and .TI
,
.
.
siiperstable .
,Xn G
in the
sense
A, then
9
U(^^\x,,...,Xn)>C^-Bn where 5, C
One
>
0.
(See Notes
and
Remarks.)
important interaction (for several stable is the Coulomb interaction. Consider
species a
of
System of
particles) which is not particles with charges
Continuous
Ci
=
e\ +
for
and
1
simplicity
h ^,z is
e2-\
zero.
Then the Coulomb interaction
r/Wfv-, 1,^1,
\
U
., ,^n)
...
^'
V"
-
/
Iv
^
l
Stahle.
is not
365
II
neutral, i.e., the total Charge
is
it
assume
Quantum Systems.
^^
_
l
v
^J\
can
that the total
ensure
Hamiltonian is stable. This is the case, for
example, charged particles satisfy Fermi statistics. (See Notes
if all the
negatively Remarks.) In
and
classical statistical mechanics the Situation is somewhat different and stabiis
lity
essentially
a
necessary and sufficient condition for the existence of the
Gibbs formalism.
f/^") be the interaction
Let
EXAMPLE 6.3.3.
energy arising from a translationally O which is upper semi-continuous. The following
invariant
two-body potential conditions are equivalent: (D is stable.
(1) (2) (3)
Z
;'^i ^y ^1 (^/
The classical
SA
=
l
^j]
-
l
>]
JA"
States äs
a
z
^^^
0, j5
>
n
> l
and all ;ci
,
.
.
.
,
.
.
jc
6
IR'
.
function
-i-Y.-^
converges for all
Stability
0 for all
>
partition
'
>
"^^"
exp{-^[/(")(xi,
.
,xn}}
0, and all bounded regions A.
of the interaction is sufficient for the existence of the local Gibbs
consequence of the results of Section 5.2 and the
following
minimax
principle. Proposition 6.3.4. Let t be a densely defined, closed, lo\ver-semi-bounded quadratic form and let T be the associated selfadjoint operator. Further, let D be a core of t and for every finite-dimensional subspace M C D define
X(M] and for every integer
m
<
l
^
t(\lj] sup ,AeM,||iA|hl
define ).,n
inf
^
/l(M)
.
dim M m
It follo\vs that Xni oo if and only if the spectrum ofT consists oo äs m of discrete eigenvalues offinite multiplicity and each eigenvahie is isolated in the Spectrum of T, In this case the eigenvalues are given in increasing order, repeated according to multiplicity, by the A,. ^
^
This is
a
(see Notes example, if and
)i.\^
and
basically Standard and Remarks). It t\ and t^
/l^^
the
are
result in operator
theory
whose
proof
we
omit
does have many interesting applications. For of the kind considered in the proposition forms two
corresponding
numbers defined
by
the minimax
principle.
366
Models of
then
Quantum Statistical Mechanics
X^^
1^^
particular if /l^^ oo. oo, then 1^^ selfadjoint operators associated with ti and ti, the condition t\ > t^ corresponds to TI > T2 and to prove that T\ has purely discrete spectrum with finite multiplicity it suffices to prove the same property for T2. In fact, more is tme. If exp{-^r2} is of trace-class for some ß > Q, then ti >
Qxp{-ßTi}
that
implies
t2
NO W if TI and T2
>
=
^ e-ß^->'' m
utilizing
>
Y, ^"^''"
<
l
-^
=
Tr(^-^^^)
.
m>\
the minimax
principle together
exp{-jS T^^^}
5.2.27 and the fact that
following
^
is of trace-class and
Tr(^-^^0 Now
and in
the
are
with
Propositions
is trace-class for all
/?
>
0
one
5.2.22 and
obtains the
result.
Proposition
6.3.5.
Let
a
U^
,
F|"\
selfadjoint
Hamiltonian constriicted
from
with relative boiind less than one, and Assume that
U^^^^
+
F^'^^
is
a
a
T^\
-bounded interaction
lower-semi-boiinded interaction
stähle, i.e., for all
n
> l
and Xi G R'
U(^^\x,,,,.,Xn]-^V^^^\x,,...,Xn]>-Bn. It
follows that the spectrum finite multiplicity and
r(n]
of H^ \ ^(T,A
A,(
>
consists
of discrete eigenvalues
with
i,(r(5)-5
1,2..., where {lf,i(X)}^^^^^ denotes the eigenvalues ofX in increasing repeated according to multiplicity. In particular exp{ß(Hff^^ i-iN\)} is of trace-class for all ß > 0 and all G [R for Fermi statistics, and all /.i < B for Böse statistics. m
=
Order and
/(
6.3.2. The Wiener The Wiener
integral,
Integral
äs we
of thermodynamic limit
shall
use
it, is
fimctions, but
a
technical tool to prove the existence origin in attempts to imderstand
it has its
Brownian motion. Brownian motion is the thermal motion of very small
particles immersed in a heat bath. If p(x, t) is the density of particles at the point X G IR^' at time /, it is well known that p is governed by the diffusion eqiiation
9p =
ä7 (in suitable units). If sufficiently tempered / > 0 given by
the initial äs
\x\
-^
^2
^^
density distribution x oo), this equation has
p(x^ 0)
-^
a
uniqiie
is
given (and is
solution for all
Continuous
p(x,t)
Quantum Systems.
j dy p(x,y',t)p(y,Q)
=
II
367
,
where
;7(^,y;0-(47rO-^'/VI'^->'l'/^^ We
the convention
use
follows that if
p(x, y] 0)
Brownian
a
=
particle
ö(x
y],
Starts at
p(x, y] t
.
where d is the Dirac a
point
measure.
It
y at time /o, then
tQ}dx
-
is the
probabiHty that the particle will be in the volume element dx near .Y at tQ. The philosophy behind the Wiener integral is that the particle in moving from y to :^ must have followed a continuous trajectory, and therefore there should be a probability measure jj,^~y^ on the set Q.x,y of continuous trajectories starting at y at time ^ and ending at x at time r, such that //^"J'^ applied to some subset K of Q.X y gives the probability that the particle has followed one of the trajectories in K. In the following paragraphs we will show that modulo the problem of giving Q.X, (or a related set) a reasonable Borel y time
t >
structure such
a measure
exists.
0 and with subBy translation invariance it is enough to consider ^ ß. For technical reasons we will assume sequent applications in mind we set t that the orbits are functions from [0, j8] into [R^ where U^' is the one-point compactification of [R^'. At the outset we put no continuity restrictions on the trajectories, i.e., we take =
=
Q^
X
=
^"
=
(^''f'^^
0
space
of trajectories, "nice
"
or
paths. (A posteriori
subsets of
the Wiener
measures
will be
Theorem
supported on 6.3.6.) Equip Q^ with the Q^, see product topology, i.e., the topology of pointwise convergence, then ^ß is a compact Hausdorff space by the Tychonoff theorem. Obviously, Q^ is so large a
space that not all Borel sets
and hence
Q^,
on
enough
to define
by
^LI'^ äs a ^|; ^, on
We first define
C(^ß)
are
Baire. We shall define
the discussion at the Radon a
beginning
yU'f
on
Baire
measure
measure.
special *-subalgebra Cfin(O/?)
of continuous functions
äs a
of Section 4.1.2, it is
Q^.
This
algebra^
of the
C*-algebra
consists of functions of
the form
Cü6%^ q)((D) =F(o}(ti},o}(t2),...,a}(t,n)) < ^i < 6 < < ^^ < ^ are fixed real numbers, and F: (Ü'')'" i-^ C is a continuous function. Thus (p is a continuous function on Q.ß by the definition of the product topology. In evaluating it is useful to notice that if
where 0
^^^y((p)
Ki,K2,
.
.
.
,
A:,
are
^The siiflfix fin is (p ^
an
Borel subsets of
U\ then
abbreviation for finite. This nomenclature is
Cfm(^ß) depends only
on a
finite number of
points (jo(ti)
on
the
apposite paths.
since each function
368
Models of Quantum Statistical Mechanics
r
r
dy l
-"l
JK,n
p(x,xi;t\)p(x\,X2]t2-ti)
JKi
p(Xm-\,Xm]
'
is the
KI
probability
that
at time ti for all /
time
The
ß,
a
=
ß
ti)dxi
-
l, 2,
.
.
.
,
f-i^^y((p)
at
x
time
at
.
^
.
=
.
dx^
0 is in the set
dy around perfectly natural.
is therefore
y at
r
/
^(^i,...,^m)j^(^,
^i)j^(^i,^2; ^2-^1)
^i;
Jr
jr
p(x,n-l,XrJ^]
t^
-
tjn-\] p(Xm, y] ß
-
t^}dxi
dx^
-
is
independent of the particular representation used for cp because depend on Xk one can use the semigroup property of to the diffusion equation,
if, for example, the Solutions
y]
and is in the volume element
Mfv(^)=/ i^i^^y((p)
tm-\] p(Xm,
particle starting
m
definition of
following
-
Brownian
r
Now
tm
F does not
p(xk-\,Xk]
JR^'
tk
tk-\)p(xk,Xk+\]
-
p(xk-\,Xk+\]
=
4'+i
tk-\)
-
4+1
tk]dxk
-
,
yuf^
to integrate out the variable Xk^ Thus is a well-defined linear functional on C'fin(^^)- Using the semigroup property just mentioned on all the variables
^1,
...
one
^Xm
immediately
I<,(
1.1^^ y
is bounded with
Weierstrass theorem
Since
ÜA^f ,^||
2.3.11
that
Q^,
on y
is
p(x,y;ß)\\F\\^
norm
and
less than
that
extends
p(x,y]ß)
yLi^^
presentation
/.(^
<
derives the estimate
implies
^f
and hence
C(O^),
<
or
Cfin(n/?)
l^i^^y(^)
=
Radon
theorem, there exists
a
P(x.y. ||
equal
by continuity
positive
a
=
is
a
to
a
p(^->y'^ ß) measure.
regulär
to
p(x,
y,
,
But the Stone-
ß).
norm-dense Radon
subalgebra
measure on
it follows from
Finally, by
of
C(Q^).
Proposition
the
Riesz
Borel measure, also denoted
re
by
such that
^^X-p)
=
/
)^<j (co)
jQfl
for all (j)
G
C(Q^).
The
measure
//f
is called the conditional Wiener ^
measure on
^<^The
qualification conditional in the definition of i^i^ stems from the spea path starting at x at time 0 shall end in y at time ß. Alternatively one could place no restriction on the path at time ß, i.e., one could ^
cification that
consider the compact space 0^
X ^' ^(t^'f^^^
,
r>o
and if cp e
C(Q)
is
a
function of the form
q)(cD)=F(cD(t^],CD(t2),...,o:>(tm]] where F fine
:
(IR^)'"
f^
C is continuous and 0
<
/^i
<
6
<
,
<
^m,
one
may de-
Continuous Quantum Systems. II
Mx(
/
=
/
"
369
F(x\,...,Xm)p(x,OC\\ti]p(x\,X2\t2-ti)
7r
Jr
p(Xm-\,Xn,]tn,
tm-\)dXidX2
-
dx^
-
This time ^x extends by continuity to a Radon probability measure on C(Q), and the associated regulär Borel measure is called a Wiener measure. If ^ is a Borel subset,
shall follow
//;^.(^)
obvious way,
is the
probability that trajectories in K.
of the
one one
clearly
particle starting
a
at
x
If /i.^ is restricted to has the connection
^x(9)
at time
C(^ß)
zero
in the
/ l^ly(9}dy
=
JR^'
for all (p e C(Q.ß). This corresponds to the fact that has to end up somewhere at time ß.
a
path starting
at
x
at
time
zero
It is well known in the
particle
is at the
elementary theory for Brownian
at the time zero, then its
motion that if
a
distance from
x at time point ^ > 0 is proportional to ^^/^. This indicates that the Wiener measures should be supported on continuous paths, and even paths which are Holder continuous of Order less than |. Recall that a mapping co : [0, CXD) i IR^' is said to be Holder continuous of order a,0 < a < l, if for all w > 0 there exists an M > 0 such x
mean
>
that
o}(s)
o}(t}\ <M\s-t'',
-
Let Q!^ be the set of such
measure on
continuous
Theorem 6.3.6 a
is
//"O
for
<
a
<
^,
ofQ.
then
a
<
and
^^(0.^)
Q^ =
is
a
measure
really
of Holder
Borel subset
paths
can
in
Q^.
be viewed
continuous
The äs
a
paths of order
ofQ.ß.
for all x e R\ If^< oc < l, then /i^(Q^) is supported by the paths co such that co(0) l
^I^M} for allx.y e R' and ß > 0, and ?/| ... ß is supported by the paths co(0) ILL^x,y
<
=
=
=
0 x.
p(^^y'^ß} l, then
a
<
x
and
A^f J^) ^y
....
p
=
0.
Furthermore,
,-.
.
co(ß]
y.
See Notes and Remarks.
The main
mechanics
the set of such
Q^
l the set Q!^
allx e IR^. Furthermore, IJL^ //"O < a
PROOF.
.
trajectories.
For 0 <
Borel subset
a
in Q and
paths
theorem states that the Wiener
following
Q<s,t<m
reason
point
for
introducing
of view, is that
one
the Wiener measure, from a statistical can derive an integral formula for the
semigroup ß^
exp{-jS(-V2
+
^)}
,
370
Models of
Quantum Statistical Mechanics
where V" is the
selfadjoint Laplacian operator on Z.-([R^') and [/ is a siifficiently well behaved real function. This formula is usually referred to äs the Feynman-Kac formula and it is a consequence of the Trotter product formula of Corollary 3.1.31. It is very convenient in making estimates on the kernel of the semigroup. Theorem 6.3.7
L"([R^').
Let
t/,
V be operator s
Assiime that U is
operators HQ bounded belovv.
=
V^,
of multiplication by real fiinctions on by the selfadjoint Laplacian
bounded
relatively
mth relative bound b <
l, and that
V ^
L\^^(R^')
is
If H
(HQ^U)
=
+
F
then^
/^Ai,(co)exp(/ l
(e-^^i^}(x)=
JQ
[
=
dt[U(o3(t])
^
Jo
dy
jr
f J^ß
V(com]i^(co(ß}} J
J/((\,Hexp{- / dt[U(cß(t})^V(cD(t))]]i^(y} ''
l
Jo
.
J
PROOF.
We begin by proving the first Statement of the theorem with U and V replaced by real bounded continuous functions and then we make a series of approximation arguments to obtain the general case. The second Statement then fol-
lows because
^(.,((p)
/
=
dyi.il^,((p)
Jr and
/if
is
supported by trajectories
^,
Let H
=
the Trotter
HQ + W where /^ is
a
such that
o}(ß)
=
y.
Consequently,
product formula
e-l>",^
(e-/'e-/'>')"V
lim
=
m
is
co
real bounded continuous function.
*
cxo
valid, Corollary 3.1.31. But by successive
e-f^^^/"\p(x)=
use
of
l dypUy-^cp(y] mj \
Jw
and
(e-^^^/'W-ß-^^^WA,^(;,) ^Strictly speaking
the
expression rß
f Jo does not make
+
y(co(t))]
for a general co e üß. Since, however, the Wiener measures are supported by paths, the expression can be defined for almost every co E O/j and the integrals formulas should be interpreted äs integrals over H where 0 < a < ^ sense
Holder continuous in the
dt[U(co(t))
Continuous
one
Quantum Systems.
II
371
obtains
((e-ß^^h^^e-ß^'-T^|J](x]=| dx,--^ [ d^^p(x^x^-^] ^J \ J
J
(
ß\
f
ß\
v
mj
\
mj
l ^ß^(^j}\,^ ^r^^^^ ^ j ^
plx,n,x^-i]-\ "'p{x2,xi',-]exp4 "2^
X
(
r
R J^
f
m
^.^j
//?A\ l
=y^^rf^,(c.)exp|-|;;r(^a)(^^jj UH^)) Now
by passing to everywhere
a
almost
subsequence
o
lim
x
assume
may
ni
/
if
V
almost all
Jo
\^JJ
(with respect
to
co
that the left-hand side converges is a continuous path
rß
/ R '\ \
^V^fcof^))- /
m-oomfr-f But for each
one
Qxp{ßH}il/. Moreover,
to
.
dtW((D(t')].
^^) paths
continuous
are
by
Theorem 6.3.6
and hence
f lim m-.oo
R J!L
/Rj\
l
(
pointwise
almost
everywhere
/ -'0'
on
W(cD(t)) ^l^(co(ß))
Q". Furthermore,
V''^/
j-\
l dn^((ü)\ijt((a(ß])\
=
el^\m\(e-^"^\^\)(x)
=
e^\\^H^Tißr'^
<
e''ll"'ll~(47r/?)-"/2( /
=
el^\m-2-^l^m<+oo
l dy e-(^-y^"l^^\il>(y)\ /
jc.
dt
c//.exp|-^X^fF(^)U(co(^)) (^ J
for all
rß
Qxp{-^yw(f^]}il^(cD(ß))=Qxp\l 7o JQ ^1 m^^ V'^/J
f
\ '/2
e-'---'')-/^/'^ ,
c/>'
Therefore, lim
/c//z,Hexp|-^fFf^U,A(W) 1^ mj^ V'"/J
'"^007
=
y"rf/i,(a;)exp|-^ fF(ß)(0)|^(a;(ß))
by the Lebesgue-dominated convergence theorem and the proof is finished for the special case of a bounded continuous interaction. Next we remove the continuity assumption. Suppose 1^ e L([R^') and that W is a sequence of continuous functions such that ^ in Wn(^]\ < ll^lloo a^^ ^n(x] W(x) pointwise almost everywhere. Then W,, the strong operator topology by the Lebesgue-dominated convergence theorem. Thus, //o + ^ is the graph limit of the sequence of operators HQ + Wn and it follows n
->
-^
372
Models of
Quantum Statistical Mechanics
from Theorem 3.1.28 that the
Q-semigroups ß > Q^-^ Qxp{-ß(HQ -\- W^)} converge exp{ ß(//o -\- W}}. Hence by passing to a subsequence one may assume that (exp{-^(//o + ^n)}'A)W converges pointwise almost everywhere to (exp{-^(//o + W)}il/)(x). But by the previous argument the Feynman-Kac formula is valid for the HQ 4- Wn and, äs Wn(x) W(x) pointwise almost everywhere it follows for almost all oj e Q.^ that Wn(cjo(t)) W(co(t)) pointwise almost everywhere in strongly
to
^
>
0
^
-^
-^
Therefore, since the Wn and W are uniformly bounded, one may apply the previous argument based on the Lebesgue-dominated convergence theorem to deduce that t.
lim
/^Ai,Hexp|-/ l
"^^Jü =
dt
'
JO
/^/^Hexpj/ l
J^
Wn(cD(t))}il,(co(ß)) J dt
JO
W(co(t)}]i^(aj(ß)}) J
and the
proof is complete for W eL'^(U"). //o-bounded U is locally square integrable because for find a i// 6 Cg^([R'') such that i//(x) l for .T G A and hence
Now each may
each A c !R^
one
=
(V^A[ dx\U(x)A
~<\\Ui^\\
\+b\\H,il
/
Therefore
one
may introduce
Un^ m(x)
It
immediately
core
follows that
a
double sequence of
approximants U^^
=
mm(U(x),n)
if
U(x)
=
max(t/(x), -m)
if
U(x)
>
m
G
I^(R^) by
0 ,
<
0
.
|| [/,, i// ?7i/^|| -^ 0 for all il/ eC^(U'). But Cg(lR') is a by the relative boundedness of t/ it is a core for HQ -^ U -{- W Consequently HQ + U,,, + W converges in the graph limit to -
of HQ and hence
where W
HQ -\- U
G
L([R').
m
-\- W and the convergence of the
corresponding semigroups follows from Theorem 3.1.28. But the Feynman-Kac formula is valid for the potentials Un^m + W and the right-hand side of this formula converges to the appropriate expression with U -\- W by the previous arguments concerning the continuous paths in Q.^. Here one uses the Lebesgue-dominated oo and convergence theorem to control the limit n ^
the monotone convergence theorem to control the limit m ^ oo. Thus by the previous argument one has the desired result for U -\- W. Now since V G ^/^^([R'') and K is lower semi-bounded, there exists a sequence Wn G 1*^(5^'') such that W,, is monotonically increasing and Wn(x) > V(x) almost everywhere. But it then follows from Lemmas 5.2.13 and 5.2.25 that the Q-semi-
ß > O^^Qxp{-ß(HQ + U + Wfi)} converge strongly to the Q-semigroup ß > Q\-^Qxp{-ß(HQ + U) 4- K)}. Thus the general Feynman-Kac formula follows by repetition of the previous arguments. groups
So far, we have developed the theory of the Wiener measure and the Feynman-Kac formula only for the infinite- volume Laplacian V~. In order to apply the theory to finite-volume Gibbs states and their thermodynamic limits, one has to develop the corresponding concepts for the finite volume Laplacians T^\- The theory for T^ ^^ follows from the theory for T^ '^ and the tensor product structure. Thus we concentrate on T^ ^. Consider the diffusion equation
Continuous Quantum
^ with
-
V^,
^ >
0,
^(^,0 ^C7(x)p(x,t), is
II
373
,
condition
boundary
(which
G A
X
Systems.
interpreted
p(x, ^)
äs
:=
0 in the
t>Q, xedA
case a
=
oo)
and with initial condition
p(x,Q)=f(x} where, for example, /
p(x,0
G
=
^^(A).
This
(e-'^-^/m
equation has ^
the
unique
l p.^^(x,y'J]f(y]dy
solution
,
J^
The function p^^^ is the kernel of the semigroup ?i-^exp{ usually called the Green function for the problem. It has the
?r^ j^}
and it is
following
prop-
erties
(1)
Positivity
conservation:
Pa^^(x,y]t] for
(2)
0,
^ >
>0
jc, ;; G A.
Semigroup property:
/
pa,\(^,y]t]pa^\(y,z]s]dy
=
pa^\(x,z]t-]-s)
.
JA
(3)
Contraction property:
/
P^.^(^^y\t]dy<
l
J\
for
(4)
with equality if and only ;c G A, / > 0, boundary conditions). Monotonicity in boundary conditions: If ö-i(jc) < 0*2 (jc) for all jc G (9A, then
P
once
again
A,
>
if
z^^ A
0
(Neumann
p^,^(x,y]t)
let =
(47rO-"/VI^--''l>
be the Green's function for the whole space and define the
by
==
^ > 0.
p(x,j;0 function
d
compensating
Green
374
Models of
Quantum Statistical Mechanics
(x, y]t)
^(T,A
p(x, y]t)-
=
p^^
A(X,
y ;
0
for jc, j; G A, / > 0. This difference z^-^ A describes how the free diffusion process is affected by the introduction of boundary conditions. One would expect
Z^^A(JC, y]t)
to be small when
corroborated
by
the
Theorem 6.3.8.
For each
kA(^,y;OI for
all x,y
A,
0
Fiirthermore, the
fiinction
>
x
following
< t <
and y are far from the result.
ß
0 there exist constants
>
C,c' ,c"
9A and this is
0 such that
>
Cp(x,y',t/c']Q^p[-c"d(y,d\]~/4t]
<
vvhere
ß,
d(y, 9A)
C, c' and c"
constants
boundary
can
is the distance
be chosen
to
be
0, and they
from y to 9A. independent of the that if R > 0, then
are independent ö/A in the sense /l>0, C>0,c>0 such that for every bounded convex domain A C [R^, whose boundary ÖA is a C^-surface of mean curvatiire less than \IR, the folloK^ng ineqiiality holds er
there exist constants
Fff,A(-^ ,y;0|
<
C^^V-^'/'exp{-(c(J(;c,^A)- J(y,ÖA)')/4^} +
.
PROOF. We will only prove this result in the simple case of Dirichlet boundary oo. In this case, if y G A is fixed, x, conditions, i.e., er /^^ZQO, A^^, JF;0 =^(^Tt) is nothing but the solution of the diffusion equation =
^-V-z, dt
jcGA,/>0,
with initial condition
z(x,0) and
boundary
=
0,
.T
G A
condition
z(x, t]
=
p(x,
y-
1), xedA,t>0
.
Hence, by the maximum principle, 0 <
z(x,t)
<
sup
p(x',y]i:)
<
0
(47iT)~'/'e-^^^>''^^)'/^^
sup
.
0
x' &0/\
But the function
T^(47ÜT)-^'/'e-^'/^^ is
increasing
with
T
when 0 0 <
for 0
< ^ <
<
T
<
d^/2v
z^,A(;c,j;;0
d~/2v. If, however,
t >
<
and hence
(47iO"^'/'e-^^-^''^^^'/''
d~/2v,
we
can
use
the
positivity
conclude that 0 < <
Zoo,ACT,j^;0
<
p(x,y]t)
(4ntr"^'< (^ntr''-e-^(y^^^^"'^^e'''^-
.
of /?CO,A to
Continuous Quantum
Hence, in all
z^,f,(x,y;t]
The last estimate in the theorem
e^/2(47rr)-''/2e-''(^'^^)V*
<
now
construct functional measures
pa^\ in almost constructed from p.
probabilities were
The
375
trajectory
exactly
.
follows because
Zoo,^(x,y\t)=z^^^(y,x]t) forall^,7eA
can
II
cases
0 <
One
Systems.
the
.
corresponding
same manner äs
the transition
to
the Wiener
measures
space will be
X Ä=(Ä)M-
ÜA.,=
0
where A is the closure of A. One next defines on
positive
a
is compact in the
Again QA,JS
Radon
measure
/^J'^'^
product topology.
C(Q./^^ß) by first defining it
on
functions (p of the form
(p(co)-F(co(^l),Co(^2),...,Co(^m)) where F
:
(Ä)'"
i-^
C is continuous and 0 < ^i < ^2 <
"
-
X
.
The Riesz theorem then
assures
.
the existence of a
=
l
-
unique regulär
^H^/^:;^' l
\
/
'
A
JD.^.B A,/? It follows from the
/ij'^'^
.
^^_i)
-
Borel
measure
such that
^^:\^y^(9] nition of
^ by
<
/'(7,A(^l,^2 ;^2 ^l) Pa,\(Xm-\,Xm',tn, dXm Pa,\(Xm ,y\ß- tm)dx\
X
QA,^
^/
JA
JA
)Uj;^'^
<
/ l F(x,,...,Xm]p,^^(x,Xl',t,]
/^'t^(^)-^
on
,
monotonicity
that the functional
,
of p^^ A in
co) ^
_y
ö"
measures are
(property (4))
and the defi-
monotonic in the
boundary
conditions, i.e., 0 < ö"! < (T2
implies
that
^oc.^A,/!(5) for all Borel sets B One
these if 0 <
< +CO
can
<
;,-^^A,^(5)
<
^'/'''(Ä)
<
/l;^'^(5)
C.Q./^^ß.
prove the
analogue of Theorem 6.3.6 for the measures /^J;^'^, i.e., supported by the set of Holder continuous paths of order a the set of Holder continuous paths of order a has measure zero
measures are a
<
^ and
ifi
ditions and how
they
physical interpretation
influence the functional
of the various measures.
boundary con boundary
Dirichlet
376
Models of
Quantum
Statistical Mechanics
give rise to Solutions p of the diffusion equation with p(x, /) boundary 9A. In the context of Brownian motion this means that a particle which hits the boundary is immediately absorbed or annihilated. This interpretation will be corroborated in a precise mathematical way in Corollary 6.3.10. In the context of heat flow, Dirichlet boundary con conditions, =
ditions heat
a
0 for
=
X
oc,
in the
that the medium outside A is at temperature zero and absorbs all cx) corresponds to completely abreaching the boundary. Therefore, 0mean
=
sorptive boundary
p(jc, t)
conditions, This is also reflected in the fact that the solution
tends to zero,
oo. uniformly in jc, äs ^ boundary conditions, er 0, give rise to Solutions p of the dif fusion equation with dp/dn 0 at the boundary (9A. In the context of Brownian motion this can be interpreted äs stating that any particle which hits the boundary is reflected into A again, with the result that the gradient of the particle density normal to the boundary is zero. For heat flow this corresponds to a completely reflecting boundary condition. The intermediary cases 0 < (T < co correspond to semi-reflecting, semi-absorbing boundary condi tions, i.e., more and more particles are absorbed at the boundary with increasing er. In the context of quantum mechanics, Dirichlet boundary conditions have a completely difl'erent interpretation, while the interpretation of Neumann boundary conditions is the same. The diflusion equation is replaced by the Schrödinger equation ^
Neumann
=
=
f =-^* on
I/^(A),
with the
appropriate boundary condition, and this equation dea noninteracting quantum-mechanical particle confined f(x) the equation has the unique solution
scribes the motion of to A. If
\lj(x, 0)
=
^(x,0 and if
=
(e-'
l, the measure x G Ai-^ |i/^(jc, ^)|"^jc describes the prob||/|| \\\l/\\ ability that the particle is in the volume element dx, located at the point x, at time t. In particular, the eigenfunctions of T^ \ describe the stationary stable States
=
of the
=
particle, i.e.,
\il/(x,t)\^
\f(x)\^.
But for Dirichlet
boundary probability of flnding the particle near the boundary of A, while the total probability of Unding the particle somewhere in A remains constant, i.e., the boundary is repulsive. If on the other hand one has Neumann boundary conditions, then there is no effect of exclusion near the boundaries. In fact, the state corresponding to the lowest eigenvalue (lowest energy) has a constant eigenfunction and there is equiprobability for flnding the particle near any given point, i.e., the boundaries are perfectly elastic. This interpretation is corroborated by examination of the boundary conditions d\l//dn (Ji/^. The parameter a is related to the elasticity of the boundaries with positive values corresponding to repulsion and negative values corresponding to attraction. For example, if er is sufliciently negative. conditions these latter states attribute
=
=
a
relatively
small
Continuous
then
T^ j^
has
with
eigenfunctions
Quantum Systems.
II
377
negative eigenvalues (negative energies) particle is dose to the boundary with
which describe bound states in which the
large probability. If these interpretations of the boundary condition are correct, then they really are conditions at the boundary, i.e., a Brownian particle moving inside A without hitting 9A should not "feel" the effect of the boundary conditions. The following theorem gives a precise mathematical Statement of this fact. Define a function ÄA on QA,^ by ^
if
fl
^
aA(co)
=
l 0 Since the Wiener
measure
all
co(0
GAforalUG
[0,jS]
<
^uf
otherwise
.
and the finite-volume Wiener
measures
^
/ij'^'^
continuous
supported by trajectories, it follows that the function A is measurable with respect to these measures (although not a Borel function). We emphasize that aA is different from a^. are
Theorem 6.3.9.
The
identity
/aAH^/x^'^'^M^ /aA(a;y//,^ JB
is
validfor
PROOF.
JB
all Borel sets B
Consider first the
C^/^j
case
B
Let K^ c A be
^A,/?.
=
such that Kn is contained in the interior of
Define functions OLK-
^ß
If
co
is
a
^
continuous
n
sequence of compact sets
a =
1,2,
...
and
U^
^
^
{0, 1} by
"-^
a^fco)^/! "^
K^+i for
^f
\0
coWe/:
foralUG[0,^]
otherwise.
trajectory
which does not touch
compact subset of A, and hence contained in lim
a/-^(co)
=
ÄT
some
.
ÖA,
then
CD
([Q,ß])
is
a
It follows that
c(.\((jo)
nCG
1^^.
;uj'()'^-almost
for and all co G ^A,^- In view of the yvergence theorem it is therefore sufficient to show that
/
^^A,/?
a^(coy//^'^'/^(a;):= '
/
Lebesgue-dominated
con-
a^(a;y^f^,H
J^ß
for all compact ÄT c A. Let d be the distance between K and ÖA; d pactness. Let n > l and
4=;^,
.
=
>
Q
by
com-
0,1,...,+1.
Define
an(co) Then
lima(a;)
We compute
=
^
<^
(^K(O))
l
if
0
otherwise.
o}(tk)
G ^
for all Ä:
-
0, l,
...
,71 + l
for almost all co, with respect to the functional
measures.
378
Models of
/
Quantum Statistical Mechanics
a,(co)^<';^'^(co)
/ dxi---dxnllp,^^(xM^Xk-^ V /L-^O
=
"'
Jü^,p
^+^ + 1
JK"
ß
dx\---dXnY\\
=
JK^
/=oV
p\
^k+\
,
^k ;
-
V
ß 2a,^\^k+\, Xk] n
+ /
f
\
ß
l dxi---dXn\\^p(:)Ck-^\,Xk]^^-A
=
/
=
a.H^A^f ,H+^
+
Rn
,
JO/; where XQ
=
x, x+i
But Rn is
=
a sum
y, and
of 2""^'
we
must
show that lim Rn
=
^
l terms of the form
/
dx^---d^nQ
,
.//:" where
Q
{0, 1,2,
is
a
of
product
factors
k+\
Z^^^A
and
n-
k
factors
p,
and
,n}. Taking moduli of the factors in Q, extending to R'' all integrals which do not involve z^^^A-factors and using the semigroup property of p, this term will be majorized by an integral containing ^+1 factors Z^^A! and at most k -^2 factors p. By the first inequality in Theorem 6.3.8 the Z^.A! factors are dominated by k ^
expressions
...
of the form
ß ( \ ^ d-(n^\] Cp[xi^,,xr^ ,,/, ,Jexp<^-c^^ r^
c'(n
Hence
using
+
which is valid for 0 < c'
l, extending the integrals counting terms one obtains
=
to all of
<
group property of p, and
.
4ß
the estimate
P(^,y;t)
l.
\])-^\
IR'', using the semi
-".(-^ ?) ^..^^m:^ [---{- ^}l"
.--,(,,.)([,.C.-..e.p(-.ä^}]"'-,).^^0,
This ends the
proof
in the
case
5
=
QA,/?-
IfB c
HA, ß is an arbitrary Borel set, there is a sequence of functions in C^^n(^A,ß) converging to the characteristic function of B, and it is sufficient to prove
/ jQAj; for (p G
Cfinl^A,^)-
c/)(co)c^A(co)^MJ:^'^(a;) "
-
/
c/^HaAH^/^f. ,,(co)
J>ß
But (p has the form
(p(a}) =F(cü(ti),...,a}(t:))
,
Continuous
where F
:
(IR')"'
first part of the
/
t^
C is
proof
continuous, and 0
<
Quantum Systems.
^i < ^2
<
^m <
ß-
379
II
It follows from the
that
(^(co)aA(co)JA^;'^'^(co)
J^^,ß
l
=
'
"
dxi...dx,F(xi,...,x^)
l
l
l
cf.^((Di)d^'^^^^((üi)
Jüt^
a.^(o}2]dll'^-^^^ ((D2) '
J^A,,2-'\ =
aAK, + iy/iJ^y-^'"(co,,+i)
'^^^,ß-'m
JA.'" X
/
aA(a;2y/^:;V''K)"-
^^^,l2-<\
l
aA(a)i )W^J'^;^> (coi )
J^A,,,
/
X
=
[
dx,...dx,nF(x,,...,Xm]
JA"-
'"l
"
(p(co)cc^(a))d^l (CD)
OCA(o},+i)d^l-'^'(cDm+\}
^^^,ß-'n, .
JÜß
One implication of Theorem 6.3.9 is that the Wiener measure /^^^^'^ corresponding to Dirichlet boundary conditions is obtained from the free measure /z^ by multiplication with the characteristic function aA of A. This confirms the physical interpretation of Dirichlet boundary conditions äs completely
absorbtive.
Corollary
For every Borel subset B C
6.3.10.
/i^V'^'^W
/ ^^((üW. ,H
=
QA,^
ß^^ all x^ j G A
/ilX5n{a;;
=
aA((ü)
=
one
1})
has
.
JB
PROOF.
Because of Theorem
inside A which touch d\ for
proof of Theorem
6.3.8
we
6.3.9., it is enough some
t G
[0,j?]
to show that the set^ of
has
/ij'^'^
demonstrated that ZQO, A(^5 JK 5
measure zero.
0
trajectories But in the
^ ^^^ all jc, >^ G A
,
^ >
0
and hence 0<
Using
j9oo,A(^,y;0
the definitions of Wiener
measures
l^:y'''^(B] for all measurable sets B C
QA,^?- Hence,
/^.f,,W
<
P(^^y'^t]
.
this leads to the estimate
<
nl,^(B}
it is =
o
enough
to show that
.
This is
a well-known general property of the Wiener process which can roughly be expressed by stating that if a point lies near a smooth (v l)-dimensional surface, then it is very improbable that a trajectory starting at the point will not cross the surface in the near future (see Notes and Remarks).
EXAMPLE 6.3.1 1
.
to consider the
case v
a parallelepiped, the Green's functions corresponding to boundary conditions can be explicitly calculated. It suffices l since the Green's functions in higher dimensions are simply
If A is
Dirichlet and Neumann
380
Models of
Quantum Statistical Mechanics
products of the functions one
in
dimension. Now for x^y
one
^
( Z/2,Z>/2)
^
1}L
and
t >
0
has
PO,
A
(^, 7 ; 0
X]
=
n
P(^^ y^2nL- /)
cxD
=
oo
^
+ n
p(x,(2n
+
l)L-y;t)
00
00
Y^
Poo^^(x,y-t)= n
p(x,y
+
^
2nL-t}-
00
=
,
-00
p(x,(2n
-
y-
t)
.
-00
This
again illustrates the nature of the boundary conditions. technically convenient when A is a parallelepiped boundary conditions, i.e., It is often
p(L/2, t) The
corresponding
=
p(-L/2, t)
to work with
^ (L/2, 0 g (-i/2, t)
periodic
=
,
Green's function is then
.
given by
00
pptT,\(x,y] t)
^
=
n
If
we
define
e :
R
h-^
2nL
-
if (2
/Oi^
(2n
_L. +
l^r
1)L
_ -
7z
H-^
-^L
<
x
<
(2n
+
^)L
<
X
<
(2/7
+
|)L
nL]t]
.
=
l
and';O/i
+
[- L/2, L/2] by X
8(x)
p(x,y
oo
=
ifOT^J__ if (272 +
^)L
v jc
=
0,l,2,...
^[-1/2, 1/2],^ by
[8'(com=8(oj(t))
,
then the relation between the Green's functions /?O,A ^^^ P is reflected in the following relations between the functional measures:
^?'.v'^w- E A<.(^'"'w) z
for all Borel subsets B C
boundary
6
n[_^/2,i/2],^.
-1(7)
A similar relation
To conclude this section
we
state the
a
monotonicity property
Theorem 6.3.12.
L^([R^).
Iioc([^')
used in
periodic
sequel and show that it immediately
Chapter
5.
Let U, V be operators
Assume that U is
that V G
be derived for
finite-volume version of the Feynman-
Kac formula which will be needed in the
implies
can
conditions.
of multiplication by real functions on r(i) \vith relative bound less than one and T^l-bounded 'fr,A'
is bounded below.
If
.^(1) /j( ^i'i-(^ri+^)+^ ^ö-,A V^ö-,A _
~
Continuous
and
jS
>
0, it follows that
Qxp{ßH^ ^}
is
an
Quantum Systems.
integral operator
on
II
381
Lr(^)
with
kernel
l d^l^^^ß(co]^^p[l dt(U(co(t]]-^V(o,m\ l JO
e-ß^l'\(^^y)=
JüA,ß In the
case
.
)
of Dirichlet conditions,
er
=
oo,
this kernel has the
simpler
re-
presentation :r(l)
e-ß"l '!A(X,>;)-
/ ^A/,\(co)aA(a))exp(/ l ^0
dt(U (cD(t]]
But
the
iJi^^y is positive and the integrand is monotonic in A.
(o^m]J
^ V
J^ß
Therefore,
.
one
has
following.
Corollary
Under the
6.3.13.
hypotheses of Theorem
6.3.12 it follows that the
mapping A is
positive and
This property
monotone
was
^,
-/><.(,, ^)
increasing.
used in the
noninteracting
case
to prove
Proposition
5.2.31.
the Feynman-Kac technique to obtain an integral relation -particle semigroups exp{-j8//^"]^} but the symmetry, or anti-symmetry, of the wave functions arising from quantum statistics causes some additional combinatorial complications. This will be examined in more One
can
also
use
for the kernels of
detail in the next section.
6.3.3. The
The
Thermodynamic Limit. I. Reduced Density Matrices
In Section 6.3.1 on
we
Fock space and
Qxp{ß(Hff^^ Therefore,
one
defined
a
class of
subsequently
/^/VA)}
Hamiltonians
is of trace-class for
jS
>
0 and /x
(A).
//^^A
sufficiently negative.
may introduce the finite-volume Gibbs states
^,,..^. ^.,A(^)where ^ G 9I
selfadjoint interacting
showed that if the interaction is stable, then
Tre,(^)(e-^(^-.^-^^U) T,.^^^^^(^-/..-^.))
In this section
we
the
begin
study
CO^'A?
'
of the
thermodynamic
limits
o/^'^(A}
^
lim A>oo
Since we
jS, /i,
and
write CÜA
cof'^(^)
.
'
usually held fixed, we will often omit the CO^'A?^ o}^'^, etc. The analysis of these limits a are
=
suffices. Thus, is based upon
382
Models of
examination of
a
Quantum Statistical Mechanics of functions p^
family
on
R^''^
[R^'", m
x
=
1,2,..., which
are
defined such that
o)!,(a\fi)-"a\fm]a(g,n]'--a(gi)) =
l
dx\... dXm dyi
X
P^(y\^''^ym\ ^l,...,^m)
These functions a
priori that the
for
a
and
we
.
,,
dy,n g\(y\)
called the rediiced
p^^ exist
äs
.,ym ;^i,
will show the
that the p^ have
an
.
.
.
-"Qm
density
(ym]fl (^\]
matrices. It is of
'"
fm (^m]
course
not clear
functions but this will
of interactions and values of
variety
~p^(y\,.
are
.
,^m)
=
coA^(^i)
thermodynamic
ß
and "
subsequently be established /L Formally one may write
al^^(xm]a^^(yni] 'a,o^(yi))
limit of the states exists
by first proving sufficiently small values of the proof is to find a convenient func-
infinite volume limit for
activity z exp{jß/i}. The strategy of the tional integral representation of the pj^ in terms of a sequence of functions p/^ which are analogues of correlation functions of classical Statistical mechanics. The pj^ are functions over trajectories joining the points jc, y, occurring in the p^, and correspond physically to the probability densities for finding the different trajectories. The second crucial observation is that the pj^ can be interpreted äs the solution of an inhomogeneous linear integral equation of the type already encountered in the discussion of spin Systems in Section 6.2.5. Therefore, one may derive an analogue of Proposition 6.2.45 which in the present context corresponds to a uniqueness Statement for the thermodynamic limit of the p^, and hence the py^, at small activity. The existence of the thermodynamic limit for the Gibbs states (D\ of a System of fermions follows immediately from the existence property for the p^, In the case of bosons it is also possible to relate the two properties by use of the theory of analytic states discussed in Section 5.2.3. The p^ directly determine =
the derivatives of the functions
sufiices to obtain suitable of
p\(y\,
from the
.
.
.
,
ym ; -^1,
a
t^-^cD^(W(tf))
priori bounds
,-^m)
integral equations
with
m.
and to relate the two limits it
the p^ which control the growth At small activities these bounds follow on
but for
positive interactions they may be directly integration representation of the py^, Thus, in the obtain some tentative global information on the Gibbs
deduced from the functional latter
case
one may We leave the discussion of the limits of the states COA to the next subsection and now concentrate on the p^.
states
CO.
Although the above strategy for establishing the thermodynamic limit may strangely indirect, it is quite natural to introduce and study the reduced density matrices p^ since it may be argued that their values are directly related to physically measurable correlation effects. One then sees that the Wiener integral plays a purely technical role in this context and it seems worthwhile to pursue a more pure operator-theoretic approach. In fact the seemingly most significant information openly exhibited by the Wiener integral is the pointwise seem
Continuous
Quantum Systems.
II
383
positivity of the semigroup kernels. This, for example, implies pointwise positivity for the p/^ with Böse statistics. This type of positivity is less accessible from the pure operator approach but apparently has many important consequences. We now enter into
introduce
some
detail of the program outHned above. First, we m-tuples (^i, .ocm] of points in A by jc^,
more
notation. We denote
and
.
m-tuples (coi, (D^) of paths in QA,^ by alternatively to denote the operator .
.
.
.
.
of^, The
,
symbol
p^ ^i^^ be used
(Tr5^(A)(e-^--'^-)))-'e-^'.--'^^) and the
integral
kernel of this operator. The
the associated kernel will be denoted
by
p^
m-particle part
m-
Thus,
we
of this operator and
have the
decomposition
00
©
PA=
w
to the
corresponding
=
PA,/
0
decomposition 00
5(A)
e L\^]l
=
m
=
Q
of Fock space. The operator p^ is called the density matrix for the The reduced density matrices p^^ may now be defined by
^(m^n)\^ Tr(pA,,+J ^
P^.m^2^ =
where Tr
is the
partial
trace
over n
particles.
in the or
H-
m
simply
unambiguously defined particles. The integral kernels of This is
p^. One
can
al,^(^m]ao,^(ym] 'ao,^(y\)]
'
,
"
o
because of the symmetry of PA,^+ PA, W will ^Iso be denoted by p^^^
^A^(^i)
problem.
=
then prove the relation
Pf^(y^\ ^'")
mentioned earlier. We
are
principally interested
PA in the
case
Problems
which
of Böse can
consider this latter PA, and
partition
or
in
Fermi
deriving integral representations of p^ and statistics, but this involves combinatorial
be avoided in the The
case
of Boltzmann statistics,
Hubert space (5(^)? function ZA for Boltzmann statistics are case.
appropriate
^(^]
=
@L\^r
so we
density
first
matrix
,
w>0
where
L^(A.)"
is the -fold
tensor
product
of
L^(A)
without any
tion, PA
R{Tj(^^ 7-1 n^ -/.!)} -exp{-;Si ZA'© ^^-K^r
=
>0 n>0
^'
and
2^=Y. >
0
-^Tr,,(^)<exp{-^(;/J'i-/i1)})
.
symmetriza-
384
Models of
Let co"^
(coi
=
with
QA,^
,
.
.
.
Quantum Statistical Mechanics and o;'"
(ßjn)
,
starting points
Define the interaction
(coi
=
,
.
.
.
,
be two families of
cbm]
x"^ and u"^ and end
U((D^)
of the
family
co""
trajectories in respectively.
and i;",
points y"^ by
rß
f/K)=
/ J?t/(")(60i(0,...,co^(/)) Jo
and
similarly t/(co'",co'^) is the interaction of the family co'"uco". With notations, the Feynman-Kac formula, Theorem 6.3.7, can be written äs
these
exp(-;ß//^"/"')(z'"M,y'"t>)
L
exp{-C/ffl)}J^:;,:)/^
cy")
,
^A" + '",/JJ
where
the notation
we use
j/44s,^'"."(<^")-
fn^^^^;^,;^ / \/-=i
\^=i
We w'^
/
=
in
v^^
/
take the
now
the
partial trace over the last n particles. This expression above, and then integrating
amounts to
putting
in
A for
over
w/
We then substitute the result into the definition of p/^ and decompose the integration over the w + trajectories of^ and cb^ in two Steps. The second Step is an integration over the m open trajectories in co"" of an
l, 2,
=
.
.
.
,
77.
intermediate functional
p/^(of^]
of these
obtained in the first step by integrating then summing over n. The result is
P^(^,y"}-
l
trajectories.
over
The function
the closed
trajectories
Pi,((a"']d^il^^^!(of]
p/^(of^]
is
in af and
,
JD.^mn where
p^K)
=
^Ä^E
A/
;fr?^ ^'J^^n.B '"^^^^'^.ß
Jo;i...Ja).z- + "exp{-[/(co^",(ö)}
,
>0
Z
=
exp{jÖ^}
is the
activity,
and
dcb,=
l d^::^^^(cok)du
.
JA
This is the desired statistics. The
integral representation of p^ ^^ the partition function ZA can be represented äs ZA=
V^/
Vn'^'-^^A In
analogy
case
^ö;i...Jco,exp{-t/(co'^)}
with classical statistical mechanics
we
of Boltzmann
.
call PA ^he correlation
fiinc-
tionals. We next turn to the now
given by
case
of Böse and Fermi statistics. The
density
matrix is
Continuous
PA
ZÄ;,P. © exp{-jS(//(;i
=
-
Quantum Systems.
/.nl)}
II
385
,
7Z>0
where Pg is the
whether
e
matrices
projection
-fl
=
now
or
onto the even or odd
-1. The
=
partial
subspace
of L-(A)"
according to density
trace formulas for the reduced
become 00
p^(x^;;-)
^m+n
r
^^N
^Z^-/,^^ ^/ du"^s^e-^^^^^(x'^V,n(yV))
,
^'
n=0
where
TT
is
a
permutation
of
00
^A.^
n
m
^'
TT
is
a
72
,
=
Q
permutation of
^
^
variables. The
n
'
7l
group in both expressions, and e^ the permutation n for fermions.
We
variables, and
r
E^/^""E^"^'''''"'^(""'^(""))
=
n
where
H-
_
=
-f l for
7i-sum
bosons,
is
over
the füll
but e^ is the
Symmetrie signature of
the
Feynman-Kac formula for the density ma trices. Now, however, complications arise from the permutations n of the endpoints y'^u" of the m -i- n trajectories in the path space integration. The path starting at a point Xk does not necessarily go to the point y^ along COA- äs previously. Instead one may go to one of the integration points w/, then to another one uj, etc., and finally end up in a point j/ which might diifer from yk. This has the effect that several elementary trajectories, by which we mean trajectories with time interval ß, may be inserted between Xk and the corresponding yi to build a composite trajectory with time interval jkß for some positive integer jk. Similarly, the elementary trajectories that are not used to build composite trajectories connecting some Xk with some yi combine together to form composite closed loops with time intervals which are integer multiples of ß. We rearrange the sum defining p^^, which is a sum over n and over the permutations of m -h /i elements, so äs to separate the integrations over the composite open trajectories and over the composite closed loops. We will omit the ensuing combinatorial ajgument, and only cite the end result.
proceed
kernel of
äs
previously by putting
Qxp{ßHl^\}
Theorem 6.3.14. Böse
or
into the above formula for the reduced
The
m-particle reduced density matrices of an interacting afinite region A with boundary conditions a can
Fermi gas enclosed in
be written
äs
follo\vs: m
PA(^^ J^^)
-
/=!
TT
where of^ intervals
=
(coi,
.
.
,CO;)
.
(
QG
U
=
a
/ ^^.^(^^ J ^A(^^^)
m
composite trajectories with
j\ß^ 72)6, ofthe m points y\^ ^y^n and e^ B\ l for fermions, n of for fermions. Also, for bosons and e relation functional p^(of^] is defined by the formula tations
-
.
.
.
.
time
The
=
.
'
-^
l
family of Jmß^ respectively. is
^
/>
E^^n E^'"'
first sum runs over all permu l for bosons, but s^ is the signature =
The
cor-
386
Models of Quantum Statistical Mechanics
00
p^K)
n
where o)'^
n
i-
/ Q^''^
-Z^-;, ^=
Jcöi
...
Jco,z^+^exp{-t/K,a;'^)}
,
(o^i, co) is a family of n composite closed hops, q and r are of elementary trajectories that constitute of^ and of respectively, ZA, is the appropriate partition function, and z exp{jß/i} the activity, Moreover, (/(co^",(:o") is the integral fr om 0 io ß of the potential energy of the =
.
.
.
,
the number
^
q +
points integrals da) r
on
the
are
+
m
trajectories co''^ co"
n
;-
00
^^-E function
is
time t modulo
ß. Finally, the
EXAMPLE 6.3.15.
=
/.
d^^::^^^^ß(co]du JA
given by
/
^A.^-E n
l
^ 7
j=\
and the partition
at
defined by
(^cüi
...Jcof"exp{-t/(co")}
.
Q
The formulas in Theorem 6.3.14
can be simplified in the case of simplication occurs because the finitevolume Wiener measure with Dirichlet boundary conditions is obtained from the infinite-volume measure by multiplication with the characteristic function of A, A see Corollary 6.3.10. Thus, in this case it is convenient to define p^ slightly differently
Dirichlet
boundary conditions,
er
=
oo.
This
than above. The formulas then become
(
m
PA(^"'; /")
=
y:
00
/.
/ ^
n E '^''"'
"
/
=
1
,-.
-
I*/'A('")
'
J
1
where
PA(CO'"J)-ZA,r^E =
/7>0
^/^^i---^^^''"AK",'Oexp{-^(co',c5")} "--^
p7-i
,
r
dw=^^ d^l^/^^,(w)du J ^
,
i=]
00
ZA,^
==
1
r
~T / n^Q^ -^
E
^^^
...ö?coz''aA(cü)exp{-^(ä;")}
,
'
df.L[ /
O^AV'^
y
is the infinite-volume Wiener measure, and ~\
m 1
In the
^
)
case
_
/ l
l ,-.
lO
if all the
trajectories
,
.
.
.
,
co^^, coi
,
.
.
.
,
of Boltzmann statistics the formulas become
pA(^'";j'"') where
coi
,1
otherwise.
=
/^A'.f,,K>A('")
,
co,
remain inside A,
Continuous
n=Z^^ Y^
PAI<^^
^ ^^
^0 o
[ düi...d(b,z'"^"y.^(of\
Quantum Systems.
387
II
ö}")Qxp{-U(o}"\ 0)")}
J
=
and
diil^^^(o^]du
dcb=
ZA
In these
cases
the
J] ^
-
/
,
J(^] ...Joj,(XAK')exp{-^(oy)}
defining
measures
the reduced
density
.
matrices p/^ from the
cor-
relation functions p^ contain no Information about the detailed properties of the System, i.e., properties of the interaction t/, the chemical potential /i, and the region A. All this information is contained in the functionals p^.
In the
EXAMPLE 6.3.16.
Dirichlet obtained
of the ideal Böse
case
or
Fermi gas, U (^, and with for the partition function =
boundary conditions, er oo, the expression in Example 6.3.15 takes the form
f ^^^ BJ~ ^-^J ZA,
exp<
=
,
V]
r^
[/^
7
f
C
/ / d^-^/^((I))dx aA(a)) AJ
and hence the infinite-volume limit of the pressure lim
^= A
exists and
''
can
=
be
computed
^
00
^^rrylog p\/\\
ZA^^
äs
A^^l; ^/ /""<"'"
<"'
^.-M|:9^//"--"4sÄ-5PSince both the finite- and infinite-volume states 5.2.28 and
density
5.2.29),
we
are quasi-free (see Propositions following expansion of the Pj\(x'":y'") reduced p^(x'^y} reduced density matrices:
have the
matrices in terms of the
pAC^'";/")-=E^^'n^A(^y;^(vy))
.
7=1
Here the
reduced
sum extends over all ??i\ permutations density matrix is given by
~p^(x-,y)
=
of
{ ji
,
3^2,
,
ym}
and the
two-point
f^s^'-^z^ /J/if,H^-AH -^
7=1
Thus for Böse statistics A
t-> p/^ is a Moreover the infinite-volume reduced
positive monotonically increasing density matrices p(x'";y'") exist for
tistics and
p(x"';y'")
=
Y^^-llp(xf,n(yj))
,
function. both sta
388
Models of
Quantum Statistical Mechanics
where
p(j; y)
lim
=
A^
p^(x ; y) -^^ '^^^^^
'/'
(4nß) ^ '"^^
= ~
'
sJ-^zJ
^\ -y:^exp<^ \
V
^
00
y v/2
(x-yY -
--^
4jß
To illustrate the
Utility of the integral representation for the reduced density first prove that for a Böse gas with positive interaction potentials these matrices are bounded by the corresponding matrices without interaction.
matrices
we
This form of the
a
priori estimate is particularly useful for linking the existence of
thermodynamic
limits of the states and the reduced
Note that if the interaction
potentials
U(m + n}^^m^ /)
U^^^\x'^}
>
+
density
matrices.
then
positive
are
f/(")(/)
.
This follows
by remarking that the difference between the two expressions is the interaction energy between the particles at the points x and the particles at the points y, i.e., it is a sum of positive potentials. Theorem 6.3.17. tials
O^^^
Consider
an
positive and<^^^^
are
volume reduced
density
G
interacting
L\^^(K'^).
Böse gas for which all the poten
Let
p^^
and
p^^
denote
j^
thefinite
matrices with interaction U and without interaction.
It follows that
P.%(^^^^)
0 <
for
all
x^, /"
G
A"\
all
ß
>
P^.,A(^^;y")
<
^ < 0 and
0,
>
(j
<
PS,A(^^/^)
0.
PROOF.
It follows from the integral representation in Theorem 6.3. 14 and the choice of Böse statistics that each term in the representation is positive. Thus, PQ ^>0 and for the first upper bound it suffices to prove
P.^A^)
0 <
<
P.,AK)
But 00
'."[AKO
=
n
and because
we assume
the
=
/.
l
/ o"'-'^
Z-\ ^
dw,...dünZ^^^^xp{-U(co-, ä")}
-y
potentials
exp{-^(co"^, cö")}
to be
positive
one
has
Qxp{-U(co"')}Qxp{-U(ä)")}
<
.
Therefore 00
p^^^(w"')<^exp{-U(co'")}Z-\ ^
/.
l -
-J
'^' 0^7=0
r,
-z^exp{-t/(co'")} But the
same
computation for
U
<
=
0
z^
/ Jräi ...rfcöz^exp{-(7(ö")}
=
.
gives
P,AK)=^
Continuous
Quantum Systems.
389
II
and this completes the proof of the upper bound. Finally, the Green's functions are monotonically decreasing functions of the boundary condition function a and hence the largest value for p^ 5. ^A occurs for the smallest value of 0-, i.e., the Neumann condition
If
one
0.
a
boundary conditions, then one may utilize the noninteracting reduced density matrices (see example
considers Dirichlet in A of the
monotonicity 6.3.16) to obtain Corollary
bound which is uniform in A and is
a
6.3.18.
Adopt
Dirichlet
boundary Itfollows that 0
the
general assumptions of Theorem
<
P^C^"-;/-)
lim
<
A
p^(= p^ ^)
\vhere
and /l < 0
valid for /i
0.
=
6.3.17 but mth
conditions.
f>l(^;f)
<
even
denotes the Dirichlet reduced
Consequently
-H'
pl(^-,}^)
,
00
matrices and
density
ß
>
Q
,
m
0 <
P^CX-; j"-)
< z-
;^ n p(^' n
-
^(y'^^
'
i=l
where
P(X
-
(27r)"'
l d^pe^P^'e-^P^l
-
.^-ßp-\-^
ze
Thus, for positive interactions and arbitrary A the Dirichlet reduced density matrices
In
positive
bounded functions.
density matrices for small values of the activity. Order to proceed it is necessary to impose conditions on the interactions [/A. fact, äs pointed out in Section 6.3.1, we need a stability condition
Next In
are we
discuss the reduced
>
U^
(or possibly
a
weaker condition
-BN^
the total
on
pressions
have derived for the reduced
verge without estimate
some
in order that the
path space density matrices will certainly Typically, this condition implies
partition
we
Hamiltonian)
function should exist. Moreover, the
finite-volume
condition of this sort.
U(o)^, co'^)
>
-(m
+
ex-
dithe
n)Bß
and hence 0 <
Thus
exp{ [/(co'", co")}
exp{-f/(co'", co")} is
integrable,
<
e^^ ('" + ")
and this
certainly
.
ensures
that all the
path space formulas we have considered in this section converge when the activity z is sufRciently small, i.e., zQxp{Bß} < 1. Next, for simplicity, variant
we assume
two-body Potential, i.e..
that [/A is defined
by
a
translationally
in
390
Models of Quantum Statistical Mechanics
C/W(^,,...,x,)=
^ l
We
place
several
auxiliary conditions
condition which is needed for technical would
probably
(A)
X 1-^
at the
an even
^.
-
Xj)
.
by
First,
we impose a continuity subsequent proofs but which finer analysis
on
reasons
prove to be redundant
is
^(xi
a
in
real function which is continuous except
possibly
origin.
The next condition is the
stabiUty
condition which
ensures
the existence of
finite-volume Gibbs states. The interaction is stable, i.e., there exists
(B)
y^ /
l
for any finite
Finally,
jc/)
-
<
\
a
.
.
.
,x/}
of
thermodynamic infinity.
constant
J\A
> -mB
/
j
in order that the
There exists
constant 5 > 0 such that
<m
family {jci,x2,
dition of decrease of
(C)
(D(jc/
_j
<j
a
dx\^(x)\ >
in [R^'.
limit shall exist,
0 such that O is
>
c
points
< +00
integrable
we
for
need
\x
a con
> c,
i.e.,
.
c
Now with all these conditions
on ^, and with Dirichlet boundary condi possible to establish the existence of the thermodynamic limit for small activity by the indirect method discussed at the beginning of the section.
tions,
it is
Theorem 6.3.19
(Ginibre). Lei (^ be a t\vo-body potential satisfying condi (C). Then there exists a Banach space ^ of sequences of 1,2,... trajectories (elementary trajectories for Boltzmann statistics, composite trajectories for quantum statistics), bounded linear operators K^, K on ^, vectors CA, C in S such that K^, K, CA, C depend linearly on the activity z, and a constant R > 0 such that the follomng are satisfied.
(A), (B), fiinctions of m tions
(i)
and =
For all
complex
z
with
\z\
one
has
\\K\\\
<
l, and hence the
equation PA
has
a
unique solution
=
CA
+
^A PA
p^ in ^, This solution is
pf,
=
given by
^K^U n>Q
and hence is
a
norm-analytic function ofzfor \z given by
0
< R.
In the
case
that
Continuous
P.ICO'")
II
391
l
.-IVz^
=
Quantum Systems.
-
'
n>Q
fäcö, Jcoi
in the
of Böse
case
p^(co'")
=
or
X
//7 ?//e
Fermi statistics, and
Z;^ ^ l >o
z^^''^^(cD^\ co") exp{-^
dä)
[dcb,
.
.
.
co^^)}
by
Ja;,z^^+'^aA o;")
^'^
exp{-U'(cLi'", co'^)}
of Boltzmann statistics, mth the appropriate definitions of case (see Example 6.3.15). complex z mth \z\ < R^ one has \\K\\ < l, and hence the
cfl^^
da) in each For all
(ii)
equation p
has the
t:
=
+
Kp
unique solution p
=
E^"^ rt
>0
in (f. This solution p
depends norm-analytically on z for \z\ < R, boundary of A recedes to infinity, PA(^'") converges to p(co'"). 77?^ convergence is uniform in z for \z\
(iii)
region. We will not
theorem
(see
give
the rather
Notes and
complicated technical details of the proof of this Remarks), but remark simply that one obtains the
equation P^
=
^^
+
Kf^p^
0. A very similar expansion about the expansion about the case t/A used in the of 0 was /?O proof Proposition 6.2.45. In fact in the present context the expansion is really about the case exp{-^/7A} l- Typically, one
by
an
case
=
=
=
sets
e-/"* and obtains results when the
fß
=
have
l
+
f^(x]
appropriate
small bounds.
One may now define the reduced density matrices pj^(^^]y^] from the correlation functions PA äs in Example 6.3.15, and one may use exactly the formulas to define p from p. Theorem 6.3.19 has the immediate same
corollary.
392
Models of
Corollary
Quantum Statistical Mechanics
6.3.20.
The rediiced
density matrices Pj\,(^^^]y^^) cind p(x"^]y"^) functions of x^^ y"^ and analytic functions of z for \z < R. Moreover, p(x^^] y^) is translationally invariant. When A oo in the sense that the boiindary recedes to infinity, p^(x^^',y^} tends to p(x^]y^^] uniformly in z for z\ < R' < R and uniformly in (x^, y") \vhen all the points in y.m^ ym Yemain in a fixed bounded region. Finally, there exists C G (0, 1) such are
bounded continuous
-^
that
|PA(^'"; j^")l
<
(N/0(i
A/RY' P^\^"; j^")
-
all \z\ < R^ where p^ denotes the reduced density matrices of the noninteracting Böse System at activity C- Moreover, for Böse and Boltzmann
for
statistics
one
has
~p^(x'"-y'")>Q By the matrices,
same one
methods used to prove convergence of the reduced density also show that truncations of these matrices have good
can
clustering properties by the expression
PA(^I,
.
.
.
.
in space. Recall that the matrices
,^m ; yi,
.
.
.
,
Jm)
=
co(fl^ (yi)
.
.
.
are
related to the state
al^(y^]ao,(x,n)
-
.
flco(^i))
co
,
where the we an
right-hand side should be interpreted in the distributional sense. If expectation of co äs in Section 5.2.3, all truncations with unequal number of a*^(y) and aa^(x) disappear because of gauge invariance. form the truncated
It is therefore natural to define truncated
p(x\
,
p(x]y}
=
pj^(x-y}
3^2)
=
PT(^I
X2 ; yi
,
,
expectation values
,
^2 ; Ji
,
yi)
+
PT(^\ ;
-^PT(XI ;y2)prfe;7i) etc., where 8=1 for bosons and
g
=
of p such that
yi
)Pr fe; yi)
,
l for fermions. Here
to zero for Boltzmann statistics. The düster
s
has to be set
equal
property then takes the form.
Theorem 6.3.21.
Let ^ be a translational-invariant two-body interaction satisfying (A), (B), and (C), and let \z\ < R^ where R is defined in Theorem 6.3.19. Then the infinite-volume truncated reduced density matrices pj are absolutely integrable functions of the difference of their arguments, i.e.,
l So far
we
dx\... dxm dyi... dy^i-i \ PT(^'^] y'^]\<
have not discussed the influence of
thermodynamic limit, Combining the path
but
only worked with space representation
boundary
+
o^
conditions
on
the
the tractable Dirichlet conditions.
of the reduced density matrices, derived in Theorem 6.3.14, with the fact that the influence of the boundary on the path space measures disappears when the boundary recedes to infinity, Theorem 6.3.8, one would expect the infinite-volume small activity reduced
Continuous
density
Quantum Systems.
II
393
matrices to exist for all classical
boundary conditions and to be equal to boundary conditions. This result, although fea-
those obtained with Dirichlet
sible, has
not been proven. Some tentative results
have, however, been obtained
for the pressure. If
Z.,^(ß,^l) is the
Tr(e-'^-^-'^^^))
^
then the local pressure is defined
partition function,
P,,A(;ß, ^)
(^A|)-'logZ,A(jS,
=
/.)
by .
NO W if ö"! > (72, then ffff],
in the
sense
of
A
^
ffff2,
A
forms. Thus, it follows from the minimax theorem
quadratic
that
Zoo,A(jS, f^} for
(T
>
0.
boundary
Z,,A()S,Ai)
<
to show that the
Thus,
<
ZO,A(^, A^)
thermodynamic limit exists for all positive independent of the particular choice it
conditions but that it is
suffices to show
|A|-'log^
hm A
-^
^0
00
,
=
0
A
and either lim
POOA
exists
-^
(jS|A|)-MogZoo,A
oo
or
hm
PO= A
*
(jS|A|)-MogZo,A
00
exists. The latter
questions of existence have been handled by operator-theorelatively satisfactory manner. Dirichlet boundary conditions it suffices to assume that the inter-
retic methods in For the
a
action is stable and
U(^+^\x^y^]
<
U^^^(x'^)-i-U^^\y'^}
.
when the distance between the sets x^^ and This condition
greater than R. It decrease
j/" is larger than some fixed value R. potentials which are attractive at distances fact be relaxed to deal with positive interactions which at infinity. The Utility of the condition is that it implies
corresponds can
sufficiently
in
fast
-^00, whenever AI
u
AI
This and similar
A
to
^
-^00,
AiuA2
^
-^00, AI^OO,
A2
C A
and the distance between AI and AI is greater than R. inequalities discussed below are all straightforward applica-
tions of the minimax theorems. The existence of PQO then follows for translation invariant interactions by the same type of argument äs in Proposition 6.2.38.
394
Models of
The
case
interaction,
Quantum Statistical Mechanics
of Neumann one
boundary conditions is quite different. partition function satisfies
If there is
no
finds that the
2'o,
^
A, uA2
-^0,
AI
(*)
2o, A2
for all AI and AI such that AI n AI 0. These inequalities are in the opposite sense to the corresponding ones with Dirichlet conditions. Thus if one hopes that they should remain valid in the presence of an interaction then it =
appears necesary to impose conditions on U^^^ which are in the opposite sense to the above. The Situation is particularly simple with positive potentials because one
then has
C/(^^ + ^)(;c", j;'") for all sets x'^ and y" and this
U^^^x'^) +^('")(j;'")
>
once
more
allows the deduction of the sub-
multiplicative property (*). Then the existence of PO can again be established for translationally invariant interactions by arguments of the type used to prove
Proposition
Neumann
case
6.2.38. But the restriction to
much less
satisfactory
positive potentials makes
than the Dirichlet
the
case.
Now let US consider the comparison of ZQ, A and ZQO, A- There have been two basic methods used for this purpose and they both have severe drawbacks. The first method is operator-theoretic and has the advantage that it works for all forms of statistics, Böse, Fermi, or Boltzmann, and for many-body interac
tions. Nevertheless, it has two
decreasing is
a
cube of side
length L,
disadvantages. It works only for positive parallelepipeds. Basically, one proves that if A/,
severe
interactions and for then
Po,A._.(j5,A^-8v/a2) Subsequently,
one
uses
obtain the existence of
incorporated
<
Poo,A,(^,Ai)
the existence of
POO(J^, /^)
and the
Po(jß, /O
<
/^o,A.(^,AO
^^^ i^s
identity POO
=
into these results if Dirichlet conditions
aries of the hard
.
continuity
^o- Hard
are
used
in jj, to be
cores can
on
the bound-
This is
accomplished by approximating the cores by an increasing sequence of positive bounded potentials and then using Lemma 5.2.13 and a dominated convergence argument to prove the above inequalities cores.
between PQ and PQOThe second method for
comparing ZQ, A and Z^^ A uses functional integraadvantage that for particles with hard cores one may incorporate an interparticle attraction, if it is mediated by a suitable two-body Potential. If there is no hard core, then the corresponding estimates have only been carried through for positive translationally invariant two-body potentials tion and has the
<|) of finite ränge, i.e., ^(X) =0 for |v^| > ^ for some Ä > 0. This second method has other disadvantages. It requires that the surface of A is smooth and
it
only
works for Böse
Boltzmann statistics.
Basically, the idea of this integral representation of ZQ, A and then split the functional integrals into two parts, the integral over trajectories which touch the surface 9A of A and trajectories which do not touch the surface. The latter trajectories give the total contribution to the Dirichlet partition function and hence if U is positive one readily obtains or
method is to write down the functional
Continuous
ßßj
00
1
P,^^(ß^^]-P^^^(ß^^]<
p|A|/rt
dx J
395
II
/>
/>
Y.
Quantum Systems.
Jü^jß
J\
4^,%^'^-^(co)[l-aA(co)]
But
4i^;:^'^'^(a))[l ^t^x,
/
(^^((b)]^z^^^(x, x-Jß]
-
X
-
ZQ^^(X, x-j'ß)
'J^^,j|
+
\ZQ^/,(X, X]jß)\
,
where z^^ \ is the compensating Green's function introduced prior to Theorem 6.3.8. Now combining the estimates on IZ^^^A! given by this latter theorem
together
with
some
duces that PO ,
methods
see
6.3.4. The
regularity assumptions on the surface of A one again de0 (for references and further details on these ^oo A and Remarks). ^
A
,
Notes
Thermodynamic
Limit. 11.
States and Green's Functions
previous subsection we studied the thermodynamic limit p of the redensity matrices p^ associated with a Gibbs state co^'^. In this subsec we exploit some of the information concerning convergence of the p^ to
In the
duced tion
obtain results CCR
on
convergence of the states, viewed
äs
states
on
the CAR
or
algebra.
The CAR
case
is
simple.
One has
cof;^(*(/i)...fl*(/X^^)...fl(^i)) =
dxi... dXm dyi... dy^ g\(yi}
"
-
gm(ym)f\ (^\)"' fm(Xm)p\(y^ \^]
and
ik/)ii
L^(IR^).
=
iK(/)ii
=
11/11
converge uniformly on compacts, then the states is the case, for example, in the small activity region by converge. But this Corollary 6.3.20.
for
/
G
The CCR
Thus, if the pj^
case
is
more
diflScult. One
can
obtain convergence of the states
from convergence of the reduced density matrices if all the states involved are analytic states in the sense of Section 5.2.3 and if the corresponding reduced
density matrices are suitably bounded. There is one example which is particularly amenable to this kind of treatment, the Böse gas with positive potentials. We will consider states co on the CCR algebra ^ over U^L^(A) which are obtained äs limit points of the Gibbs states COA with Dirichlet boundary conditions. The bounds for the reduced density matrices, 0 < p^ < p^ obtained in Corollary 6.3.18 will play an important role. Other boundary conditions could also be handled by appealing to similar bounds in Theorem 6.3.17.
396
Models of
Quantum Statistical Mechanics
These bounds imply that the limit points co are locally normal states over the quasi-local algebra (^, |J^^+(A)). In fact, all the states co describe a finite number of particles per unit volume. To see this we first remark that
a;AJ/)a^,(/))
j dxdy ~p^(x'^ y)f(x)f(y}
^
l dxdy~pl(x'^y]\f(x]\\f(y]\
<
j
l dxdyp(x-y]\f(x]\\f(y)\
<
where
p(x]y)
is the reduced
density
,
matrix of the infinite- volume Gibbs states
co^ of the ideal gas (see Example 6.3.16). In particular, the left-hand side is well defined. Thus, we can define the number functional for COA restricted to any +(AO) with AO C A,
A^AO(<^A)
sup^cüA(a;^(/)a^^(/))
=
5
,
/eg
where the supremum is over all finite orthonormal subsets of fore, calculating with the above bounds
O
/
dxpl(x',x)<
we
can
Section 5.2.3
dxp(x-x)
estimate the number functional of we
pointed
N^,(CD^)
=
There-
.
^AO
JAo
Now
j
L-(Ao).
out that the number
co
restricted to
^+(Ao).
In
functional is lower semi-con-
tinuous and hence
0<^AoH= A^Ao limco^ v
a
limyVA(tt;Aj
<
<
A^A^K)
.
a
This shows that the number of
hence
we
particles in AQ in each such state co is finite and have established the last Statement of the following theorem for
positive interactions. Theorem 6.3.22.
Consider
Potentials ^^^^
positive and
lume Gibbs Further let
are
states
{COA}
for be
some a
an
^
interacting Böse
>
^^^"^
G
0,/i
<
^/^^([R'').
gas
for
which all the
Let COA denote the
k-body finite vo
0, and Dirichlet boiindary conditions.
subnet such that the limits
(D(A)
=
limcoA^(^) a
existforallA GUA^+(^)It follovvs that co defines ^
=
an
entire
analytic
state over
the CCR
{^^^^(\)and lim
coA,(a)^,,^^ (g\]'-- ^OM, (On]]
-
o}(^co(g\]
'
^o^(gn))
algebra
Continuous
for allg\," ,gn ^L^(W] mth compact support and all n finite local particle density and hence is locally normal.
> l.
'
Remark.
II
Quantum Systems. The
s täte CD
397
has
The
assumption that the potentials are positive can be relaxed by the introduction of a suitable notion of superstabihty (see Example 6.3.2). Assume the interaction t/^'"^ is defined by a translationally in variant, real, even, two-body interaction potential ^ e IjQ^,([R^), somewhat
t/('")(xi,...,xj=
^ l
for X
=
U^^'^ is superstable in the
that
Suppose
(x\^
.
.
.
,
jc,)
X contained in the
regulär, i.e.,
t/(^)(^)
>
where 5 >
0,
^(Xi-Xj]
.
that
^ n(X,r)^-Bm
C
C > 0 and
is the number of
n(X^ r)
cube centred at
r. half-open a positive decreasing function \l/
points of strongly
Assume also that O is
there exists
f dtil/(t)t"-^^-^
such that
on
(0, CXD)
an
estimate
JJQ G
for
some
1/2
^ >
and
(D(x) for all
jc
G [R\ If
v <
3,
or
if O is
-lAd^l)
>
positive,
coA(exp{ö!A^;^})
one can
prove
c(diam(X),a)
<
for any a > 0, any bounded region A and all X c \ where NX is the local number operator for X. From this one can deduce the conclusions of Theorem 6.3.22. One can also deduce the modularity of the limiting states äs in Theorem 6.3.31. A real
(See
Notes and
Remarks.)
principal ingredient analysis.
Lemma 6.3.23.
Let
functionsfrom f. Assume that the uniformly in a, i.e.,
IR into
in the
{f^}
proof is the following straightforward
be
C, and
net
a
derivatives
of f^
sup a
l 2, for m differentiable, =
,
.
.
.
,
/ e
77, and all Ä: > 0.
=
0, l,.
l times
f^
continuously differentiable pointmse to a function
converges
up to order
|/j'")(0|<
n are
+
bounded
on
compacts,
cx)
Itfollo\vs thatfis
n
l times
continuously
and
f(m] w
>
that
sup [ -A', k]
Ja.
for
of n
assume
result of
..,/!
l, where the
.
f(ni]
J
convergence is
uniform
on
compacts.
398
Models of
If
PROOF.
n
Quantum Statistical Mechanics l
=
,
convergence of
/^
it suffices to establish to
Let
/.
>
e
0. Choose
uniformity points t\, .
\^[t,-h,ti
+
on
.
.
,
compact intervals / for the
t^ e / and
an
/z > 0 such that
h]=I
/-l
and
//sup|/;(OI<e/4. re/
But for
\s\
one
\Mti
+
s]
has -
fß(ti
+
s]\
<
\Mt,^s]
Ut,]\
-
+
\Mt,]
-
fß(t,]\
^\fß(ti]-fß(ti^s)\
+
tel
\Mti)~fß(ti)\
<8/2^\Mt^)-fß(t,)\
.
Consequently,
!/,(?, +s)- f(t,
^)|
+
<
e/2
\f,(ti)
+
f(t^)\
-
and uniform convergence on / follows from pointwise convergence. Now the proof proceeds by induction and it suffices to consider the because
higher
Given
t e
obtained
cases are
by replacing / by
one
case n
=
2
of its derivatives.
(R choose M > 0 such that
\/:!(t+h)\<M for all
a
and all
\h\
< 1.
By Taylor's theorem
Mt
+
h)=f,(t)+f^(t)h
+
R(t + h)
,
where
\R(t + h)\ <Mh'/2 and hence
\m-fi'M\<\Mt + h)-fß(t + h)\/\h\ for
\h\
< l.
Taylor's
It follows that
theorem
f^(t)
has
a
limit
+
äs a
\Mt}-fß(t)\/\h +M\h ^
oo.
But another
application
of
convergence
by
gives
\(Mt + h)-f,^(t))/\h\-f^(t)\<Mh/2 Thus
/
is
differentiable, and
/;(o
-
/'W
The uniform convergence on compacts follows from the the same reasoning äs in the case 72 l.
pointwise
=
The
proof of Theorem 6.3.22 follows from boimds of Corollary 6.3.18 and analyticity of the infinite-volume Gibbs state co^ of the non-inter-
the entire
acting
Böse gas,
Continuous
Quantum Systems.
II
399
(''(^(/))=exp|-cü"(c&o(/)2)/2| expj (/, (l +ze''^')(1 ze''^V/)/4| ^
-
-
(see Proposition 5.2.29). PROOF
OF
and co^
THEOREM 6.3.22.
First, choose
A(a,.(ffr) when
n
g G
^^(A).
note that the states COA
Next,
all gauge invariant and hence
are
is odd.
If, however,
A(
=
=
2m is
0
=
Co''(o(3r)
even
=
But in the last
n
=
expansion
+
-
)/2-
COA vanishes on all terms which contain an
unequal
number
of creation operators a*^^(g) and annihilation operators acoA(^)- This is again a consequence of gauge invariance, i.e., conservation of particle number. Thus, by use of the commutation relations
^<^M<,(9} one
obtains
a
<,(9}a^,(9}
=
Ibll'
+
,
representation m
a>A(
=
^q'A(3)V(9)*) k
=
Q
^
k=0
where the coefficients the reduced
density
are
/ rf/rf/ p^(/; /) n g(x<}9(^}
C^
=
positive polynomials in ||öf||^. (We have used p^ to corresponding to COA-) But by Corollary 6.3.18
p^
are
the reduced
interacting equilibrium 0
<
denote
matrices
0
,
/=!
density
state
co^(^,,,(gf'")
<
matrices associated with the infinite-volume
co^. Thus since Q'
>
0
non-
has
one
E c;r -^/ dx^dy" P(^; /) H \9(^'^\ l^ö^l
^ =
,
=
0
/=!
(l^lf").
c." (
The last evaluation follows from the gauge invariance of co^ and the definition of the f~<m
^k
Next introduce the
family
of functions
Mt) These functions converge
=
pointwise
/(O
/;
IRt>- C
cü^^(W(tg))
.
to the function
=
w(W(tg})
/;
[R
f-^
C defined
by
.
Furthermore, they satisfy the hypotheses of Lemma 6.3.23 for all
n.
For
example
400
Models of
Quantum Statistical Mechanics
l/W W l'
Hence
defined
/ e C(IR) by
and
/"'coA.(,Jg)"'fF(//))|-
=
<|tOA.(*.,(0)"")l<(1'co(lffl)"") /i"'' /<"'' uniformly compacts. on
^
F,(t) then FO is entire
Thus, / is
entire
an
=
w(W(t\g\))
^
C is
,
and
analytic
|/W(0)|
=
But if FO; IR
lim|/W(0)|
co(<5>,A\9\)"')
<
=
1^^(0)1
function. Therefore,
analytic
t^o}(W(tg)) is
entire
an
an
analytic function analytic state.
entire
for all g
^
Next from convergence of the derivatives
A.(^a...(^)") The
more
obtained
/oc(^i
,
.
.
.
,
rvi"Ho)
-
with compact support and hence
L~(M.^}
-^
one
Statement of the theorem with
in
similar
^)
a =
manner
a}/^^(W(t\g\
+ t2g2 +
by
n
examining
4-
^ö'))
is
has
rv^'^Ho)
general
co
=
cD(^o.(gT}
different functions g\, .g^ is derivatives of a function .
.
.
of several variables. We omit the
details.
There is
a
Corollary
of Theorem 6.3.22.
converse
6.3.24.
Adopt the assumptions of Theorem 6.3.22 but fiirther asdensity matrices p^ associated with the Gibbs states COA pointmse convergent. It follows that the states are weakly* convergent.
siime
are
partial
that the rediiced
PROOF.
It follows from the bounds
0
Lebesgue-dominated
L^(R^')
convergence theorem that coA(O^^^(öf) with compact support. Moreover, the estimates
'")
converges for
0,A\d\)-'") imply
that
one can
define numbers
<^(w(g)}
=
Efe!|fAii?i'"A('i'.(3)''") m
for each such g. It is then entire
Remark.
analytic
state
=
readily
CD over
0 ^
*^
'
verified that these numbers
the CCR
algebra
We have shown that the local
and COA
^
uniquely
determine
an
<^-
particle density of the interacting by the corresponding density of the noninteracting state (D^. It is not surprising that such a bound exists since for ß.i^i fixed the introduction of a positive interaction, i.e., an interparticle repulsion, shoiüd state
co
is bounded
Continuous
reduce the 0
<
p^
In
density.
p^
<
fact, the bound is just
the reduced
on
Quantum Systems.
implication
one
II
401
of the bounds
with and without interaction. This follows
density
because
[ dxp^(x',x)< JAl dxp\x]x]=N^(cD^]
N^(o})=
.
J^
But the
Gibbs states co^
noninteracting
number operator and the
0 <
can
/
J/
P^(/;/)
/
<
JA^
<
dx^
~P\:^\of]
JA^
to bounds
correspond 0
be extended to all powers of the
inequalities
(ü(N^(N^ -\)---(N^-k+\])< co(N^(Nf, -l}---(N^-k+l))
Finally,
combinatorial
a
calculation
establishes
that
there
are
.
positive
coefficients^^ S^ such that m
p'"^^S^'p(p-l)-.-(p-k+l) k=\
for all p
0
[R. Therefore
<
o)(N^]
=
^S'^CD(N^(N^
_
1)
.
.
.
(A^^
_
A: +
1))
k=i m
<
Y^ S'^(o^(N^(Nf, -l)...(N^-k+l})
=
.o/'A^w^ CO{N: A>
k=l
Theorem 6.3.22 states properties of the thermodynamic limit with ß > Q and yu < 0 fixed. It is also possible to consider the thermodynamic limit with /? > 0 and the overall particle density fixed. Recall that this latter form of limit was
essential for the discussion of Bose-Einstein condensation of the ideal
Böse gas in Section 5.2.5. The basic difRculty for the ideal gas with Dirichlet boundary conditions was that the formalism was only defined for
Qxp{ßf^} < l and for z fixed in this ränge the local particle densities p^(ß,z] were uniformly bounded, whenever v > 3, by the density of the infinite
z
=
ideal gas at
z
=
l, i.e.,
P^(ß.^] (This
was
discussed
<
prior
Pc(ß)
to
=
(2nr
Theorem
f d^p(e^P'
5.2.30).
-
1)-
Now it follows from the bound
A^A(<^) Pc(ß], one must have ZA > l for each finite A ^^The S'^ recurrence
are
known
äs
5'^""^^
=
relation
Stirling numbers of the l and S'" S'^_^ + kSl\ =
second kind. Their =
1.
positivity
follows from the
402
Models of
and in the
Quantum Statistical Mechanics limit at fixed
thermodynamic
this leads to
density
a
ränge of choices
ZA such that ZA ^ l The discussion of the interacting System at fixed density is less tractable and it is not evident that the corresponding ZA would converge to .
a
critical finite value. Nevertheless, the above method of bounds could be used discuss
properties of any Hmit for which ZA One may also obtain results for the small which are not necessarily positive. to
Theorem 6.3.25.
local Gibbs
Consider
<^^'A)
o}\(=
states
<
<
z
.
for interactions
activity region
Böse gas
interacting
an
Theorem 6.3.19. For 0
sumptions of
l
^
satisfying
all the
as-
R, it follows that the limit s of the
^^^^f,
0}(A}
lim
=
a)\(A)
A^co
for
all A G
UA ^+(^}^
This State is entire
lim A^
The
COA(
L^(W]
G
proof is
6.3.24. It relies
.
.
.
coA(^ajA(^)")
on
quantities
are zero
((/,)
the CCR
algebra.
0, and for
(D^(a^(g]^a^^(g)^ )
g e
l
=
L^(^g)
=
one
l
d^
^Aj-
proofs which
if
n
of Theorem 6.3.22 and
are
obtained
is odd. But if
X^C>A^(.7/a,(3f
=
A-
>
state over
and
with compact support,
(., (9)"")
A
Cf
=
almost identical to the
bounds
on
gauge invariance these
with
unique
a
CXD
for ///!,..., fn PROOF.
^^^ determine
analytic, locally normal,
äs =
Corollary by
follows. First,
2m,
)
0
has
J/p^(/;/)n ^^(J/) i=\
A^,
<(z/Q(\-z/RY^jJ^, dJ^j^^, J/ //(/;/) n |g(x,)|bb.-)l
,
z- l
where
we
richlet
have used the bounds of A
boundary conditions,
ample 6.3.16, A
and hence
we
(g)*aco. (
-^
p^^()C]y]
replace
may <
(z/C)(l
where co^'^ is the Gibbs state of the 0 < and the
Under
proof
coA(
now
slightly
more
Corollary
p^^^
a
-z/Ä)-'coO-H
noninteracting
(z/C)(l
<
6.3.20. Now, since
we are using Dipointwise increasing function, Exby its thermodynamic limit p^'^\ Thus,
is
Böse gas at
activity C- Therefore,
-z/Ä)-'a;.f(4.o,c(|3|)-"')
previously.
äs
stringent conditions on the interaction time-dependent Green functions
it is also
to prove convergence of the
GA(^O,
-
-
-
,^; ^1,
.
,
.
.
,
^;0
=^
A(^O
T;^(^I)
'^^(^))
:
possible
Continuous
Quantum Systems.
II
403
where
T^M) but this is
substantially
O is
(A) (B) (C)
O w/z/c/7
a
difficult to prove. Böse
=
O(x))
Fermi gas
or
the
satisfies
(^(x)
even
g^'^(^A-/iA^A)^g-/?(//A-/iA^A)
more
Consider
Theorem 6.3.26.
body Potential
=
interacting through
t\vo-
a
requirements and continuous
for
:^
^ 0,
the interaction is stable,
OGlHr)nL-(r).
Assume Dirichlet
boundary
GA(^-;r)
-
conditions and
define
local Green's functions
Z;^iTr.^^(A)(^i^-(^-^^>)^^ X
A^e-^^^-^^^^^
.
.
.
^(C".-^ViXA^^^^^-(/J+G-u)/:A)^
where
ZA //ere y4'"
=
Tr5^(A) (e-^^^ )
=
(y4i,
.
.
.
denotes
,y4,)
AI
=
^
A(fi)
or
^A
C"
=
(Ci
5
5
Cw)
denotes
/^oo, A
fl*(y/)
in the Fermi in the Böse
-
mth
case
case
m-tuple of complex
an
A^A^A
-
m-tuple of operators
an
^(fi) and
=
,
numbers mth
C^
G X)
where D
=
{r; ReCi
<
ReC2
<
-
<
ReC.
<
ReCi
+
^}
-
Itfollowsfor sujficiently small values ofthe activity z exp{^^i} that functions G(.4'";C'") analytic for C" G T) a^zöf continuous for C
exist
there G
X
^wc/z that
G(A"^- r")
lim
=
GA(/("'; T")
,
A>00
where the convergence is
uniform
on
compacts in X).
We will not attempt to prove this result although some of the necessary estimates will be derived in the subsequent discussion of quantum gases (see also Notes and Remarks).
Although
the Green's functions
G^(A,B',t]
=
o,^(Ae^^^-Be-^^^-]
converge in the small
activity region one would not expect this result in general. example, phase transitions occur, one would expect the GA to have several distinct limit points each giving information about the distinct phases. In fact, the bounds For
if
\G^(A,B-t]\< \\A\\\\B\\
404
Models of
imply
that there
Quantum Statistical Mechanics exist subnets
always
G(A,B]t)
such that the limits
A
\\mG/^^(A,B]t]
=
exist for all observables
noffs theorem. G.
Clearly,
A, B, and all t e U. This is a consequence of TychoThus, it is of interest to have an interpretation of the functions
the values
a;(^)-G(^J;0) determine have
an
a
state
the observable
co over
automorphism
group
T
of
^,
G(A,B',t) It is however too
optimistic
to
algebra of
or
=
expect
and
71(^(^1)",
w(Ar,(B)}
ideally
one
would like to
such that
.
find this structure under all circum-
to
Nevertheless, one can derive weaker forms of this result. But before doing this we single out some relevant properties of the G\(A,B; t) which are mostly preserved in the limits. stances.
The definition
G^(A,B-t}=w^(Ai^(B)) immediately implies (1) (2) (3) (4) (5) If
A
.
that
G^(A,B] t) is linear, B^G/^(A,B]t) is linear, t G\(A,B] t) is continuous, GA(^,C5;0) CDf,(ACB) GA(^C,5;0), L GA(1J;0)-coA(l) ^
-^
-
-
=
{^/}|<-.<-,j
C
^ and
{^/}i
finite sequences,
C [R are
one
has
n
Y,G^(A],AJtJ-t,] ',/=! n
=
Y^ (Uo,M^co,(Ai)^cn^,U,,^(tj)7l^^(Aj)^^J ^7=1
^U^,(ti}na,,(A^)ü,
CÜA
i=\
and hence
(6)
E GA(^*,^,-;O-^/)>O. ^7=1
From the fact that COA is
(7)
There exists
a
ß
a
T^- KMS
state at value
ß
one
also has
eU such that
r dtf(t]G^(A,B',t)^ r dtf(t^iß)G^(B,A',-t] J oo
J oo
Continuous
Quantum Systems.
II
405
A,B,e^ and all / with /e ^. (See Proposition 5.3.12.) Of these properties, (1), (2), (4), (5), and (6) are clearly preserved under weak limits. This is not necessarily the case of the continuity property of condition (3) but it for all
would follow if
one
had bounds
the derivatives
on
uniform both in A and t. This would also
\dG/^(A,B] t)/dt\
imply uniformity
which
in t of the
are
con-
vergence G\(A^B] t) -^ G(A^B] t) and hence the weak KMS condition (7) would follow in the limit. We will discuss these bounds later, after first investigating
the
structure of
general
6.3.27.
Theorem
a
Lei
G;Mx^x[Ri-^CZ?^
(1) (2) (3) (4) (5) (6)
function G
be
^
a
satisfying (l)-(7). with
C*-algebra
a
identity H,
and
G(A,B] t) ts linear for all B in ^ and t in [R, G(A,B] t) is linear for all Ae^ and t G R, t G(A,B\ t) is continuous for all A,B e^i, G(A,CB]Q) G(AC,B',Q) for fl//^,5,C, e 91, G(1J;0)-1, E"j= l ^(^;,^y; 0 ^/) > 0 fa^ allfinite sequences {^/}i <,<,
A
^
B
^
let
such that
function
\-^
=
-
C
^,
ßöfOJi<,
Itfollows
A^o}(A) is
a
State
on
2l. Let
There exists
(^,710;, ^o))
=
be the
G(^,A',Q) corresponding cyclic representation. a strongly continuous
Hilbert space ft containing ^, and unitary representation U ofR on S{ such that ft
(i) (ii) for
=
a
V.eff?^(0co
G(A,B] t)
=
(7r^(^*)Qa;, U(t)nco(B)Üa,)
all A^Bj
9l, ^ G IR. Furthermore, properties (i) and (ii) determine (5^, U) unitary equivalence in the sense that if (^ U'} is another pair satisfying (i) and (ii), then there exists a unitary operator W : S{ ^-^ S{' such that up to
^
^k
=
is.
and U (t]
for
=
W''U'(t)W
all t G IR.
PROOF.
Condition
(6) applied
on
the trivial sequences
{^4}
and
{0} implies
that
G(yl*,^;0)>0 and hence
(4) implies
co(^*^) Since
co(1! )
State. Let
G(1] H ; 0) ((y,7reü,Qöj) =^
,
=
l
=
,
G(1,^*^;0)
it follows
-
>
0
.
corollary of Proposition representation of 9l.
äs a
be the associated
G(^*,^;0)
2.3.11 that
co
is
a
406
Models of
Condition
implies
Quantum Statistical Mechanics
(6), applied
to the sequences
/ G(A, ^*;0)
and
{^,0},
where ci,C2 G
C,
G(A, ß-t) G(B\B-0}
(G(B-\A^--t) is
{c\A*,C2B}
that the matrix
positive.
It follows that
G(5*,yi*;-0
=
G(A,B-t)
and
\G(A,B',t)\'
o}(AA*)oj(B*B)
<
Hence the function G defines on
7if^(^)Q(,j
X
7ioj(^l)Qw
G
co
a
one-parameter family
^^, i.e.,
on
(7ü^(yi*)Q,Jr, n,,(B)Ü,,)
=
implies
11^/11 The definition of
.
one-parameter family of bounded sesquilinear forms
a
G(A,B-t) on
||7i,^(^*)Q,,f \\n,,(B)^4~
The closures of these forms define
t\-^Xt of bounded operators
and the bound
=
<
l
implies ^0- l
and condition
(6)
has
äs a
consequence that
X^(^,,^,^._,iA,.)>o u=^
for all finite
[t]
sequences{i/^^}j< .^^^
in
^.
Now let R be the vector space of all mappings cp from U into ^ such that the set t e [R, (p(t) ^ 0} is finite, and convert A into a pre-Hilbert space by defining
=^((P\(t)^^s-t92(s)}
(91^92}
s, t
This is
last
really a positive semi-definite inner product by the inequality derived in the Paragraph. In analogy with the procedure employed in the beginning of Sub-
section
2.3.3, the GNS construction,
we
define
3-{c/)GR;((p,(p)=0} and 5\
l//
G
f,j
äs
the Hubert space
completion
is identified with the element (p e
,
r lA
.
^^^ ^
^
R/3
-
0
it follows that
{(p,(p) and this identifies If / G [R, define
,^ a
äs a
map
subspace of of ^ by
Ü(t)
in the induced inner
defined
if ? 0, otherwise,
^ ~
\
of
=
5\.
(\lf,\lj]
by
product.
If
Continuous
(Ü(t}cp}(s) Then
Ü(t)Ü(-t]
Ü(-t]Ü(t]
=
((Pi Ü(t]q>^}
=
,
l,
cp(s-t)
II
407
.
and
^((pi (s],X,,_, cp,(u s,
=
-^
=
Quantum Systems.
-
t]]
u
Y.^9i (s' + 0,^'+i-(.'+0^2(0) s', u'
=
Y^(9i(s^t),X,_,(p2(u))
(Ü(-t)(p^,(p2)
=
.
Sjll
It follows that
lifts to
U(t)
unitary operator U(t) on R, and t^-^ U(t) becomes a unitary group on R. This homomorphism is weakly continuous since t\-^X( is weakly continuous by assumption (3), and
homomorphism
a
of !R into the
(^1, ^w^2>
-
w,^-. y^(9i Z/^ s,
=
^
0)
u
6^^ is
a strongly continuous unitary representation of [R (weak equivalent by Corollary 3.1.8). identification of ^^ äs a subspace of 5^ immediately implies t>
continuity The
-
Yl((p\(s)^^i^+t-s(p2M} s,
Hence
^2("
u
and strong
are
(i)
ft
V
=
^'Ö
teU
and, furthermore,
G(A,B',t) where (pi,(p2 ^
^
=
(7la,(A'')üa,,X,7lo,(B)Q,,)
defined
^TQ
=
((P^,(P2)
,
by
/^a>(^*)n. ^,N ^^^^~\0
if^
-
0
,
,,
if^T^O
and
(,^|^cn(B}^co
ifs=t,
^2^'^-|o ,,
But then (p2
=
Ü(t)(p^,
if.;^
r
if5
0,
.
where
,(,^^|^<^(B}^^
^3(^'-\0
=
if.T^O.
Hence
G(A,B-t)
=
((p,,Ü(t)(p2)
=
(7r^(^*)0, /7(07r,Wa.)
,
where the last step uses the identification of ^ äs a subspace of 5l. It remains to prove that (R, U) is unique up to unitary equivalence. Let be another pair satisfying (i) and (ii), and define a map ^ : 5l i^ 5l' by
W
l ^
^ U(ti)7ia,(Ai}m^ /
/
=
^ U'(ti)7i^(Ai)üa, /
This map is well defined and isometric because
.
(5l', U')
408
Models of
Quantum Statistical Mechanics
(U'(t]7i^(A]^^, U'(s)Ti,,(B]^^)
and W extends
by closure
to
=
(7c^(^)a u' (s
=
G(A\B-s-t]
=
(U(t}n,,(A)Ü^,U(s}n,,(B}Q,)
isometry
an
V
t)n^(B)ü,)
5\. But W is onto since
on
U'(t)f)^
-
ft'
=
t^R
and
one
has that
^iö. One then verifies
in the
äs
6.3.28.
(7)
together
\vith
=
2.3.16 that
W'^U\t)W
.
be a C'-algebra with identity H, function satisfying assiimptions (l)-(6) of the foUo\\nng weak KMS condition:
Let
G:^x'5Kx lRi-^CZ?e 6.3.27
^ö.
proof of Theorem U (t]
Theroem
=
There exists
a
"^
a
and
let
Theorem
U such that
ß
noo
/oo dtf(t]G(A,B-t]= OO
for allA.B^^ Let rem
and
(^,^, 7Cf,j,Q(o, 9*?, U]
/
dtf(t
+
iß)G(B,A--t]
J oo
auf with /
G
^
.
be the representatives associated with G
by
Theo
6.3,21,
follows that Ooj is separating for 7if^(^]" Furthermore, the positive selfadjoint operator U(iß) is a form extension of the modular operator A associated mth the pair (7if^(^l)'',Ocü), i-^It
,
||(7(//?)'/Vll
=
l|A'/-.A||
i.e., if
a
unitary
/oA-a//i/^eD(A'/-). Warning.
Even if Ä
,,
=
G(A,B-t] it does not follow that constructed
äs
group
on
f)^,
such that
(Q,7i{^)[/(r)7r(5)Q)
=
U(iß]
t/ is
and A
are
follows: Let 301 1 and 9Jl2 be
equal.
von
A
counterexample algebras on a
Neumann
can
be
Hubert
with a common separating and cyclic vector Q such that 901 1^9012 (such pairs are common in relativistic quantum field theory, see Notes and Remarks), and let AI and A2 be the associated modiilar operators. If ^ G 9Jli,
space
we
have
||A;/2^Q||
=
||^*Q||
=
||Af ^Q||
Continuous Quantum Systems. II
409
äs SRi is a core for A/ it follows that A2 is a form extension of AI. But A2 7^ AI by the following reasoning: If A2 AI and J\,J2 are the modular ^ conjugations associated with 9[)li and yjl2^ then J^l^-l'^ is an extension of JiA^
and
=
and hence Ji
J\. But then
9Ki and this contradicts
we
have
JiSn^Ji ^Ji^^Ji
=
^i^^2-
But in this
G(A,B-t] with ^
(Q,
=
=
J2^l2J2
case
^A~'''%Q)
^[Ri satisfies the axioms (l)-(7), and
=
,PROOF.
=
,
Q,,
-
The first part of the theorem is
9J12
=
Q,
one
^(0
has the identifications -
A-^'/^ .
proved exactly
äs
in
Example
5.3.13.
Next let
U(t] be the
spectral decomposition of
=
j dE(p]e-^P^
U. Then
assumption (7) implies
that
J d(AÜ^,E(p)AQ^)f(p) J d(A^Q^,E(p)A^Q,)f(p)eßP =
for A G
nco(^)"
\\AQ^f
and
one
concludes that
(^Q.^Qa.)
=
=
J d(A^Q^,E(p}A^^,,)eß^
=
\\U(iß}^^^A^^4^
,
i.e.,
||A'/2^*nlp The last conclusion of the theorem
||t/(;;ß)'/2^*Qf
=
now
follows because
.
naj(^)"^cü
is
a core
for
A^/^.
We next examine criteria for the limit Green's functions to satisfy (l)-(7) of the proceeding theorems. As we remarked i t is only the
(3), which
properties continuity in in a semi-abstract setting mechanics. Subsequently
a problem. We will work typical models in statistical we discuss some applications. Let (^, {^IA}) be a quasi-local C*-algebra, and assume that each ^A acts äs an irreducible algebra on some Hubert space ^. We say that a family ^A ^ ^A of*-algebras is an isotonic family if
t, property
which is devised to
poses
cover
(1)
each
(A is
(2)
if AI
CA2, thenC^A, ^^^^
an
irreducible
subalgebra
In Order to define Gibbs states
Each //A is
a
we
of
introduce
^(^) containing
a
family
the
identity,
of Hamiltonians //A-
lower-semi-bounded^^ selfadjoint operator
on
^,
with the
property that
ZA-Tr^J^-/^^A)<^_^ ^'
Note that
we
have
now
implicitly
assumed that the chemical
corporated in //A and hence lower semi-boundedness corresponds
to
potential stability.
term
liN/^ is in-
410
Models of
for all
^
>
introduced
Quantum Statistical Mechanics
0. The
family
of Gibbs states COA associated with the System
are
then
by
coA(^)=Tr,(e-^^M)/ZA where A e ^^. The finite-volume Green's functions GA
GA(^,
are
B-t}=i:T^^(e-f^^^Ae^^^^Be^^^^}/Z^
where A,B e^^ and
t G
spectively GA) äs A Green's function) on
oo can
U. Note that each
defined
by
,
limit
pointwise point of COA (reby continuity to a state (respectively ^ U^ ^A, and by a slight abuse of language we refer to such limits äs weak* limits of COA (respectively, GA). (See the discussion in the beginning of Section 6.2.2.) In the following we will encounter commutator expressions of the form [H/^,A] and the additional assumptions we now introduce are designed to ensure that these expressions have a well-defined meaning for a large enough set -^
of ^. We
(H)
=
assume
There exists
if ^ G
(1) (2)
be extended
an
isotonic
family DA ^ ^A such
X)A, then AD(H^}
C
that
Z)(//A);
the closures of the operators
[H^,A]e-ß^exist and
of trace-class for all ^ G DA and all
are
a;A([//A,^]*[^A,4)
(3) where CA is
independent
<
Q
< +
00
jö
>
0;
,
of A and the bound is valid for all A G DA
and all A. Note that condition on
D(H\).
(1) implies that the operator of condition (2) is defined (3) the definition of COA is extended to unbounded the appropriate operators are of trace-class.
In condition
operators for which
Let {GA} denote Proposition 6.3.29. the family of Hamiltonians {H\}, and
the Green's
functions associated with point of the GA- Assume that there exists an isotonic family {DA} of^-algebras satisfying (H), and assume that 7rco(UA^A) i^ weakly dense in Tico(^)" where
co(A]
=
G(^,'ll;0)
let G be any weak* -limit
is the state associated with G.
It follows that G
satisfies:
(1),(2) A,B^G(A,B] t) is bilinear for all t G R; t ^ G(A,B; t) is continuous for all A, B e"^; (3) (4) G(A, CB-, 0) G(AC,B', 0) for all A,B,C e ^; (5) G(1J;0)-1; (6) E/,y= l G(AlAji tj /,-) > 0 for all finite sequences {^/}i <,<, C ^, ^^{^/}i
-
=
and all
/
G
^.
Continuous
Before
proving
the
proposition
we
state
an
Quantum Systems.
immediate
II
411
corollary.
Corollary 6.3.30. Lei {COA} denote the Gibbs states associated with the family of Hamiltonians {H\} and let CD denote a limit point of the COA in the weak* topology. Assume there exists an isotonic family {T)A} of*-algebras satisfying (H), such that 7i(ü([J/^ I^A) ^-^ weakly dense in 7i(^(^]" It follows that co is a modular state over ^, i.e., the cyclic vector Q^; associated mth co and 71^) is separating for 7Caj(^)''. .
PROOF
OF
COROLLARY 6.3.30.
Aa such that GA.^, converges
just is
co, and G satisfies
now a
PROOF
Assume that
to
some
co lim COA,, and let Ay be a subnet of function G. The state associated with G is then
properties (l)-(7) of Theorems
6.3.27 and 6.3.28. The
corollary
consequence of Theorem 6.3.28.
OF
PROPOSITION 6.3.29.
Properties (1), (2), (4), (5), corresponding properties for GA. a net converging to G.
and
(6)
are
obvious
consequences of the
Next let
be
GA
OBSERVATION 1.
formly for PROOF.
Assume
G\(A,B\t]
If ^ e ^ and 5 e
UA ^^
then
G^^(A,B',t]
G(A,B:t]
uni-
(H) imply
that
-^
in compacts.
t
that ^ e 31
and 5 e
UA -^A-
The
conditions
is differentiable in t and
-^G^(A,B,t]
=
"H'~\iH. Rn^-''"'A|2 \cOf,(Ae''"'^[iH/B])e
=
co^(e-""^Ae""^[iH^,B])f
dt
<
Observation l is Observation l
now a
w/,(AA')w^([H^,BnH^,B])
consequence of
pointwise
<
\\AfCB
.
convergence and Lemma 6.3.23.
implies that
t>^G(A,B-t) is Continuous for /l e 9l and 5 in
sequence B
UA ^A
e
UA ^A-
Now if 5 is
a
general
element in
21,
such that
7i(5)Q
^
T(5)n
.
It follows that
ft)((5-5)*(5-5))-^0 and
we
have
\G(A,B;ti)-G(A,B;t2)\ <
|G(^,S-5;f,)| +
<
+
|G(^,5;A)
-
G(A,B;t2)\
\G(A,B-B;t2)\
2co{AA*y'^co({B-B)'(B-B)y'^
+
\G(A,B;tt)
-
G(A,B;t2)\
choose
a
412
Models of
Quantum Statistical Mechanics
for all .4 G ^ and ^i , 6
(R. The
G
of
continuity
G(A,B-t)
t^
for
general A,B ^^ immediately follows. This completes the proof of (3). If / G ^ C 5^, then / G 5^, and it follows from Observation l and the KMS property of GA^ that /co
/oo dtf(t)G(A,B-t]= -00
for
^,^
UA ^A-
G
As
Ti^^dJA '^A)
dense in
by Theorem 2.4.1 1, U^ DA such that
is
/
dtf(t + iß)G(B,A--t)
J -OG
weakly dense in 7ia;(^l), TCoXUA ^A) is strongly*ifA.B G ^ are given there exist sequences A^.Bn
and hence
(ß((An-AY(An-A))-^Q
co((An-A)(An-AY)^Q Oj((Bn-BY(Bn-B))^(^ co((Bn-B)(Bn-B)-^)^0
,
,
,
.
It then follows from estimates of the type
|G(C,AOI'
G(A,,Bn-t)-^G(A,B-t) in
uniformly
Hence
t.
rOC
/OC dtf(t)G(A,B-t)= 00
for all
^,5,
*!H. This finishes the
G
/
dtf(t + iß)G(B,A--t]
J 00
proof of property (7) and of the proposition.
Remark.
If, under the assumptions of Proposition 6.3.29, A is a net such that GA^ converges pointwise to G, and if, furthermore, 5^ ^^ where 5^ is defined in Theorem 6.3.27, i.e., there exists a unitary group U on ^ such that =
G(A,B-t]
=
(^^,no,(Ä}U(t}n^(B)ü,,)
,
then the limits
G(A,Bi,...,B,n',ti,...,tnr)
limTr^^^ (^-/^^A.^^//,//A^^^/(^2-^,)//A.^2
-
^^^-''^'"^^^J/ZA^
a
exist for all ^4 1,
G(A^B{,
.
.
=
.
5l,
...
,5,; ^1,
.
,5,
.
.
,
G
^, and
one
tfn)
(ü^,7i,,(A)U(t^)no,(B^)U(t2
To establish these Statements choose
limits 6.3.27
G(A,BI, one can
Tif^ of ^ and
a
.
.
.
,5,; ^i,
.
.
.
,
^w)
group
Ü
-
ti)n,,(B2) "7i,,(B,,)U(-t,n}^^) subnet
on
a
^
.
A/ of A such that all the By an extension of Theorem Hubert space A D )^^, a representation a
exist for all
show that there exists
unitary
has
m.
such that
Continuous
Quantum Systems.
II
413
G(^,5i,...,5^;^i,...,^^0 -
(Q^,,,n^(Ä)Ü(ti)7L^(Bi)Ü(t2
If P is the
orthogonal projection
from ft onto
U (t]
and since U is
unitary Ü
^,
one
.
deduces that
PÜ(t]P
=
must commute with P. Since
n^(B] and Qto is
ti]n^(B2) "n^(B^]Ü(-t,n)^^]
-
cyclic for (TT, 17)
Pn^(B]P
=
it follows that
A
=
^^
and hence
G(A,B\,...,B,n]t\,.. .,tm] =
(Q^,7l,,(^)t/(A)7reo(5l)^(^2-^l)7Ca>(52)--7l,,(5,)^(-^,0^c.)
This shows that all weak*-limit
itself converges. We end this section with the
previous
of the net
points
applications specific models.
some
pages to various
are
equal, and hence the
of the formalism
developed
net
on
6.3.4.1.
Quantum Spin Systems with Long-Range Interactions. We adopt the assumptions from Section 6.2.1 for spin Systems, and put ^A for all finite subsets A c L. Let ^ be an interaction such that
notation and
DA
=
^||cD(^)||<+oo X^x
for each
x
^L.li follows from
d(A)
Proposition lim
=
A>
exists for all ^ G
UA ^A,
6.2.3 that the derivation
i[H^(\],A]
oo
where
H^(\)
=
^
(D(jr)
.
XC A
The finite-volume Gibbs states
are
Trg,(e-'"^*(^)^)
"^^^^=T|:M^ ,^^
for ^
^A. In this
case
the commutators
[^a)(A),^] bounded in
A
since the limit
CXD exists, and assumption (H) is particular, Proposition 6.3.29 and Corollary 6.3.30 apply, and each weak*-limit point CD of the net{coA} is faithful on na)(&]" In Theorem 6.2.4 it was proved that if the interaction O satisfies
are
trivially
norm äs
^
satisfied. In
^e^''(sup ^|<1)W||) ;
^0
v-^,fe,
<+^
414
Models of
for
Quantum Statistical Mechanics
0, then the closure of the derivation ö generates a group of automorphisms of ^ U^ ^IA. There exist some interactions > which do not this but nevertheless the weak* limits co^ condition, CD exist, and ö^ satisfy of of a Notes and Remarks to generates automorphisms 7^to(^)^^ (See group Section 6.2.1) A
some
>
=
-^
6.3.4.2.
Böse Gas with
Repulsive Interactions.
In Section 6.3.4
we
obtained
information about the Böse gas with repulsive interactions by a method priori estimates on the reduced density matrices. This information can be
some
of
a
in certain
complemented
Theorem 6.3.31.
by
cases
Let ^
=
the
exploitation
UA^-(A)
be the CCR
and consider the finite-voliime Dirichlet
temperatiire ß
through (1)
>
boiindary
0 and chemical
two-body potential
a
^
/i < 0
potential satisfy ing:
algebra
over
condition Gibbs
for
a
UA^"(A),
states COA at
Böse gas
interacting
positive, i,e.,
)(x) for
of commutator estimates.
>
0
all xe r
77?^ estimate
(2)
[ d^'x(l^ x^)^+Xx)|-<+oo /
is
Let
CD
It
validfor
0.
be any weak*-limit point
follows
(i) (ii)
some 8 >
of CD\
äs
A
^
oo.
that
co
is
locally normal,
co
is
a
modular state, i.e., Q^co is separating for
Ti(j^(^]"
,
It has already been proved for more general repulsive interactions that co locally normal, see Theorem 6.3.22. In the Remark following that theorem was
established that
co
m
>
0 and all finite
present theorem is
it
satisfies local number estimates of the form
(ß(N'J^) for all
is
regions
to combine these
<-foo
A. The strategy behind the estimates with estimates on
||[roo,A,4^ll
proof
of the
ll[^ci>,4^ll
,
for suitable .4 G ^_ (A) in terms of
II^A-AII for suitable 6.3.29.
m
>
0. One
can
then
verify
the conditions
(H)
used in
Proposition
Continuous
Quantum Systems.
II
415
We prove the commutator estimates in a series of lemmas. These lemmas are stated both for Böse and Fermi gases with later applications to low density gases in
denotes
mind, a
see
Section 6.3.4.3. We work
on
Fock space and \l/ always D(N'J!^) where A and m
vector in Fock space which is contained in
Vary from lemma to lemma.
Lemma 6.3.32.
Iff^L"(\),
and
L^(^'},
then
\\[U^,a(f]m< |^||2lK/)(A^A-A'- 1)^11
,
\vhere m
Q
=
^^
=
for fermions
for
^
bosons
and A- A'-
PROOF.
Using
{jcG [R';;c-;^-z,>;G A,zG A'}
.
the action of annihilation operators and of ^D,
(a(/)iA)^"H;ci,
.
(U^^^Jt\x,,.
.
.
.
.
,x,0
-
,^)
=
(^1
l)'/'
+
^ l
l
cD(x,
^x
7W'A^"^^^(^,^i,
-^,.)^(")(^i,
.
.
.
.
.
.
,^)
,z,)
-^^5^a)(x/-^y)^A^"H^i,...,^.)
,
ij^J one
calculates'^
ll[C/$,(/)]iAf =E("+') r X
[dx,---dx j dx dyf(x)f(y) Y,^(x
-^
n>0
-^
now sum
the
l
n
right-hand
is chosen to contain
5]
side
(
+
''J
,dx^---dx
::::"'
to think in terms of the
U^=-
a(x)
i)/rf^E/ J -^
basis of L-(A') which
f(x)f(x-Xi+Xj) ^'"+^^(x,X^,...,X,:)ll^^"-^^^(x-X,+Xj,Xl,...,X)
'~At this point it is convenient
where
complete orthonormal gives the inequality
over a
This
,iro X
.
J
O/||O||9.
\\[u^,a(fm\ '< 110)11?
and
a*(;c)
are
x>)\J
\Y,^(y-yi)U("+^\x,x,,...,x^^^^^^ Ly=i
We
-
\_i=\
defined in
dx
dy^(x
Example
-
expression
y)a''(x)a''(y)a(y)a(x)
5.2.1.
.
416
Models of
Applying
the
Quantum Statistical Mechanics
Cauchy-Schwarz inequality
||[t/*,a(/)],A||-
<
=
Finally, for fermions
then finds
ml^Yl -^/..^_v dx,---dx\(a(f)^^)^"\x^,...,x)\>0
=
one
^.6A-A'
/,7
||I.||^||]V^_^-a(/),A||ml\\a(f)(N^_^.-im^
.
has
one
*(/)(/)
ml
<
1 ,
and for bosons
*(/)(/)
Lemma 6.3.33.
///
||[t/*,*(/)]^||
PROOF.
e
I-(A)
II/II2
<
and O e
L-(A')
<
||'ll2ll/ll2l|A^A-A"AII
<
ll>ll2ll/ll2ll(A^A-A'
First note that the Fock space
^A
with 0 e
A',
then
for fermions
+
1)'^X-A"AII
^^.^([R'') decomposes
J?(A-A005((A-AT)
/Or
bosons.
äs
,
(A A')"" denotes the complement of A A' and 0 denotes the symmetrized anti-symmetrized tensor product according to the choice of statistics. As all operators occurring in the Statement of the lemma act on ^] it is enough to 5(^ prove the inequalities on this space. Since [U<^,a(f]] annihilates the zero- and one-particle spaces,
where
-
or
"~
[U^,a(f]](N^_^-^)-^ is
uniquely
defined
äs an
operator
A'). By
^^(^
on
ll[t/,(/)](^A-A'-ir'll in the fermion
case.
Since A
C A
-
A^
we
<
for bosons.
Taking adjoints n -\- 1-particle
space into the
and
using
space
we
one
has
Il/ll2llll2
have A^A
||[f/
Lemma 6.3.32
<
^A-A'
and
<
||/||2||
the fact that
[L/(D,a*(/)]
maps the
/i-particle
obtain
\\lU^,a'(f)]N-[^.\\
<
II/IMIOII,
for fermions and
||[t/*,a*(/)]yV^-!^,(WA-A'+l)'^'ll for bosons. These
Next for bosons
inequalities we
are
equivalent
to
<
those stated in the lemma.
consider the field operators
Il/ll2ll
=2-'/2(a(/) +*(/))
Continuous Quantum Systems. II
and
Weyl operators
W(f} äs
417
-
e^'^(^)
in Section 5.2.1.
Lemma 6.3.34.
If f
e
L^(A)
L^(/i'}
A',
then
independent o//, ^,
and
and ^ G
with 0 G
\[U^.W(fm\<\m,\\fUao^ where ÜQ and a\
PROOF.
positive
are
constants
ij/.
One has
r(/)*[/7vp, W(f}]
W(fYU^W(f)
=
[
i
=
-
U^
dsW(sfY[U^,^(f)]W(sf)
.
JQ Therefore
\\[U^.W(f)]i^\\
l
<
ds\\[U^,^(f]]W(sfm
.
Jo We
now use
^(/)
(a(f)
=
+
fl*(/))/\/2
and Lemmas 6.3.32 and 6.3.33 to obtain
C ds\\(N^_^.+^f^w(sfm
\\[u^.w(fm
.
JQ But if g e
L^(A),
has
one
^(^)*A^A-A'^(^)
-
A^A-A'
=
=
i[JQ
ds
W(sgY[N^-^>,^(g)]W(sg)
[ dsW(sgY^(ig)]W(sg)
Jo
f^
=
=
where the next to last step used
f^(^/r(AfA-A'
+
/ JQ
ds{^(ig)-lm(-sg,ig)^
^(ig)
Proposition
l)'^'M^(^/)
=
(A'A-A'
+
\\g\\ly2
,
5.2.4. Therefore,
+
^1'(;/)
+.^11/11^ 1/2 +
l)'^'
and thus
\\iu^,iv(fm <
v^mi^ 11/11,
f'ds JQ
Now,
we use
(N^_f,.+s<S>(if)+s^\\f \\ly2 + ^f^ ^
/
the commutation relation
[N^-^',^(g}]
=
-i^(ig)
Models of Quantum Statistical Mechanics
418
valid for g
G
L-^(^) together
with the bounds
^(g)-<2(N^
(D(^)
(A^A_A'
^
+
!)'/- M,
ijj
G
D(N'J^)
Let
i//
g^(L-(lR''))
e
+
]\[U^,
m9\\l
H)^
in terms of
^(/)]'AII
be such
ivhere
m
=
t hat
l
f
for fermions, G L^(Af) for
m
+
some e >
It
+1)^
Inserting
every A G U''
for
one
has
i)%,.
=^ for
bosons, and c,//^,^ is arid take ^ such that
A/
some
1= f d''x(l^\x-Y^^\^(x}\" for
(A^A_A'
the lemma follows.
and
ii(^A+irvii<(iAi Flirther let
,
2^/^(AAA_A^+t)^/-||^||,
<
s(^(if) +5-||/||2l]/2
this estimate in the last estimate for
Lemma 6.3.35.
+
independent of
A.
<^oc
0.
follows that
\\[U^,a#(fM\
a or
||[t/T, w(fm\ where ao and a\
PROOF.
are
<
and d is
*
a
constant.
Further, for bosons,
^(c,A,,; A/; )lll^||LII/ll2(o
+ l
\\f\\2)
constants.
Partition IR'' into
a
cubic lattice with unit cells
A/, and defme ^/
=
x\,^-
One has
\\[U^,a*(f)m
<Ell/il2ll't'.-|l2ll(A^A/-A,+')"'l/'ll /
+!)"
/
where the second step is
l A/ -A/1
+ 1 is
equal
to
an a
application constant
df
of Lemmas 6.3.32 and 6.3.33. Next note that
and
II.
>
inf
(l
/ii'p/iii
,
where üi
=
X
e
+
2^v+e
x\-Y
A/
Thus
Eii'*''ii2
^
iii^LZr'^'
Continuous
Combining
these estimates
one
Quantum Systems.
II
419
has
ii[^T,*(/)]'Aii
/
^
The second Statement of the lemma follows from Lemmas 6.3.32 and 6.3.33
by
an
identical argument.
Next
we
turn to
estimates
involving T
Lemma 6.3.36.
// /
G
dr(-v^}
=
^^(A)
the kinetic energy operator
is tmce
.
continuously differentiable,
||[r,a(/)],AII
one
has
,
,
\\^here m
m
PROOF.
0
=
for fermions^ for
-
bosons.
One has
([T,a(f]]^l>)^''\x,,.
.
,x)
.
=
=
-
jdx /WV>("+')(^,^i,
.
.
.
,^n)
-(a(VV)'A)<"'(^i,.--,^.)
and hence
[r,a(/)j
The first estimate in the lemma follows the first
äs
-a(VV)
=
.
immediately,
and the second is
proved
from
in Lemma 6.3.34.
Lemma 6.3.37.
Lei
f
G
^^(A)
be
tmce
continuously differentiable.
It follows
that
II[^,^(/WII PROOF. One argues
äs
<
in the
\\[T.w(fm\<
l|V/||^||iAII+2||VVll2ll(A^A + l)'/Vll
proof
l
of Lemma 6.3.34 that
ds\mf),T\w(sfm
Jo =
r ds\\^(iv^f)w(sfm Jo
=
/
ds\\{^(iV^f)+sRe(f,V^f)1}^\\
/
c/5||{0(/VV)+^l|V/||^1}./'||
Jo =
JO
,
.
420
Models of
Quantum Statistical Mechanics
where the next to last Step
uses
Proposition
5.2.4. One then
(I.(/VV)'<2||VVll2(A^A + 1)
uses
,
etc.
PROOF and
OF
by
THEOREM 6.3.31.
the
subsequent
The limit state is
Remark
locally normally by Theorem 6.3.22,
has local number estimates
one
COA((7VAo+ir)
C.(|Ao|+ir
<
whenever AQ C A where c, is a constant independent of AQ and A. We will apply Proposition 6.3.29 and take DA äs the linear span of all W(f) where / is twice continuously differentiable and has compact support inside A. We have
^A
^oo,A
=
^O
+
-
/^A^A
Suppose that/is twice continuously differentiable region AQ. Then
[rco,A, m/)] for all A
b\(f)
is
independent
we
one
the
[T, w(f]]
=
6,(/)a;A((7VA+1))<6|(/)c,(|Ao|+1)
Similarly
from Lemma 6.3.35
one
has
^(cf ;Ao;)-|||
<
,
+
I)^)
<
C3(|Ao|
+
l)'
.
has
[^^,W(f)\ by
open
used
A((/VA Finally
<
of A.
A([WD,r(/)j*[Wt,fF(/)]) where
some
AO, and hence
D
A([7'oo,A,r(/)r[ro,,A,r(/)]) where
and has support inside
reasoning
used in the
W(f){(if)+ {\\f\\ll}
=
proof
\N^,w(f}nN^,w(f]]
=
<
of Lemma 6.3.34, and hence
(^(if]+\\\f\\i\rmf)+mi\] Ö2(/)(yVA
+
i)
and
o)([N^,W(f]]*[N^,W(f)\) Combining
62(/)ci(|Ao +1)
<
the estimates of the commutators of
TQO.A, ^CD,
.
and A^A with
W(f)
we
conclude that
COA([//A, ^(/)1*[^A, W(f)]] where
b(f)
is
independent
normal it follows that 6.3.30
implies
that
co
of A.
a
is
weakly
and
< +(X)
,
dense in
7rcü(|JA ^(^))'^
co
^^^
is locally Corollary
modular state.
6.3.4.3. Dilute Böse and Fermi Gases.
Dirichlet
b(f]
Hence, condition (H) is satisfied. Since
Tia^(\Jj^ X^A)
is
<
Theorem 6.3.26 stated convergence of small activity region,
boundary condition Green's functions in the because of uniformity of the convergence the limit
Green's functions
are
Continuous
Quantum Systems.
II
421
one may apply this theorem directly to conclude that modular state without verifying conditions (H). It is however
continuous in time, and the limit state is
a
interesting to note that condition (H) is actually used in the proof of the uniformity in Theorem 6.3.26, i.e., one first uses low-activity expansions to establish pointwise convergence of the Green's functions G/^(A"^',C^} for C^ G T), and then uses estimates on the derivatives of GA with respect to ,Cm? to prove that the convergence is uniform on compacts. These esti Ci, mates are of the form (H), where T) A is taken to be polynomials in a(f} and fl*(/) with / twice continuously diiferentiable functions in L^(A.) in the fermion .
.
.
case. This assures that the commutation with the kinetic energy part of //A is a bounded operator, and the bounds on the commutator with the potential en ergy part are established by rather straightforward, but long, estimates in
which the condition O 6 L^
DA
plays
an
essential part. For bosons one may take W(f) where feL^(A.) is twice con
to be the linear combinations of
tinuously differentiable.
6.4. Conclusion
Throughout this volume we have tried to demonstrate the Utility of operator algebras and their states in the analysis of quantum statistical mechanics. The most detailed and interesting justification of these techniques is provided by the theory of quantum spin Systems discussed in the first half of this chapter. A large class of these models, including all the basic examples such äs the Heisenberg and Ising models, can be described by C*-dynamical Systems (^, T) and States of these Systems correspond to the physical states described by the model. A global viewpoint of this type is essential if one desires understanding of such basic questions äs the nature of thermodynamic phases, mixture properties of the phases, etc., and this is perhaps the greatest single advantage of the algebraic methods. Traditionally, equilibrium states had been described by a variety of methods, e.g., implicit or exphcit thermodynamic limits of the Gibbs ensembles, the principle of maximum entropy, etc., but in all these methods the affine properties of the states were unclear. Phase transitions were partially understood in terms of nondifferentiability of the thermodynamic functions, or through lack of clustering of the states, but no framework really existed for the definition and characterization of pure phases and mixed pha ses. The realization that the equilibrium states could in fact be identified äs states over the appropriate C*-dynamical Systems immediately provided this framework. The equilibrium states at each fixed temperature were seen to form a convex set with the extremal points corresponding to pure phases and the mixed states to mixtures of phases. This immediately motivated much of the analysis of decomposition theory, invariant states, periodic states, almostperiodic states, etc., described in Chapter 4. This point of view was initially somewhat speculative but the theory of spin Systems provides a clear vindication of its validity; all the traditional definitions of equilibrium states coincide and they are indeed described by convex subsets of the state space of the spin algebra ^. The global analysis then reveals many properties which were hitherto unclear, e.g., the links between purity and clustering, factoriality, and mixing. Many of these properties are consequences of some form of asymptotic abehanness. The second
phenomena
striking feature in the algebraic description of equilibrium played by the KMS condition. Starting from the Gibbs
is the role
ensemble it is evident that this condition is satisfied but it is that this condition alone should characterize equilibrium.
completely unclear Nevertheless, this is
Conclusion
423
large class of quantum spin Systems, and also for the ideal Fermi surprising result is both of practical and conceptual Utility. For example we argued in Section 6.2.5 that under a variety of circumstances there exists a unique KMS state for the inünite spin System. But since the thermodynamic limits of Gibbs states are automatically KMS states, this immediately gives a strong uniqueness Statement for the thermodynamic limit, e.g., this limit must be independent of boundary conditions. Moreover, uniqueness implies that the equilibrium state is a factor state and one can immediately deduce spatial clustering properties. On the other hand, we have seen in Chapter 5 that the KMS condition has a variety of characterizations which emphasize different physical features such äs stability under perturbations and ergodicity in the form of asymptotic abehanness. This clarifies to a large extent the nature of the equilibrium states even if it does not provide any profound explanation for their definition. Since the spin Systems are described by C*-dynamical Systems, one can define the time development of arbitrary states and in principle one could tackle the problems of approach to equilibrium but work in this directhe
case
for
a
gas. This rather
tion
seems
difficult and much remains to be done.
is doubtful
description of continuous Systems is notably incomplete. But applied to their analysis and it that any complete understanding of these models will be obtained
in the
future.
Finally, this is
a
the
feature of all methods which have been
near
Notes and Remarks
Sections 6.1 and 6.2.1 The
of spin Systems äs models of magnetism predates the füll developquantum mechanics. The earliest model of this type was suggested by
study
ment of
Lenz
[Len 1] in 1920 äs a possible microscopic explanation for the phenomenon magnetization. Lenz argued that atoms in a crystal appeared to have a number of preferred orientations and that the potential energy of an atom with respect to its neighbor should favor their alignment and hence give rise to a spontaneous magnetization. Lenz emphasized the simplest case in which an atom has two possible orientations, the spin-:^ model of Example 6.2.1. The eigenfunctions of 0-3 with eigen value -f l are traditionally taken to represent the positive magnetization, spin up, and the eigenfunctions with eigenvalue -l then correspond to negative magnetization, or spin down. The potential energy, or interaction, suggested by Lenz takes the form -Jö-gcr^' for "neighboring" points x,y ^ L where / > 0. Thus, the energy is negative if, and only if, the atoms at x and y have the same orientation, i.e., the spins are parallel. In 1925 Ising, a Student of Lenz, carried out an exact calculation of the partition function, Tr(e~^^), for the one-dimensional I Z Lenz model [Isi 1], of
=
in his doctoral dissertation.
Ising inferred
dimensional model failed to exhibit
an
from his calculations that the
ordered
ferromagnetic
one-
state at any
temperature but he also argued incorrectly that the
prevail
in
higher dimensions
same Situation would and backed this argument with various approx-
imate calculations. Since 1925 the Lenz model has almost universally been referred to äs the Ising model (see Example 6.2.2). A füll history of this model to 1966 has been given by Brush [Bru 1]. Ising's negative conclusions appear to have been accepted by other authors and in 1928 Heisenberg [Hei 1] proposed a more complex interatomic spin-spin coupling, Ja^ a-^, äs the root of ferromagnetism (see Example 6.2.2). While the Lenz-Ising model is basically a classical model, i.e., the Hamiltonian is a function of the mutually commuting third components of the atomic {0-3}.^^^ spin, the more sophisticated Heisenberg model is inherently quantum mechanical and in this sense is the prototype of quantum spin Systems äs we describe them. Nevertheless, the Heisenberg interaction is not necessary for the understanding of ferromagnetism because in 1935 Peierls [Pei 1] proved that
Notes and Remarks
425
Lenz-Ising model exhibits a spontaneous magnetization in two or more J^ and v > 2. (For a further discussion see the Notes dimensions, i.e., with L and Remarks to Section 6.2.6) Spin Systems on regulär lattices also provide a model of crystalline structure. It was a long-standing notion that crystals were formed in a repeating pattern best described by a periodic structure. This tenet of symmetry then imposed a number of restraints on the possible crystalline forms, e.g., rotational symmetries of order greater than five are impossible for periodic patterns in two or three dimensions (see, for example, [[Sen 1]], Chapter 1). But the 1984 experiments of Schectman, Blech, Gratias and Cahn [Sehe 1] revealed a metallic phase of an aluminium-manganese alloy with a six-fold symmetry, the first example of a nonperiodic crystalline structure. These experiments radically aifected ideas about crystals, their structure and definition. For example, the definition of a crystal äs proposed by a 1992 Commission of the International Union of Crystallography is 'a solid with an essentially discrete diffraction diagram with the symmetry of the crystal that implied by the diffraction pat tern'. Thus the traditional definition of a crystal has been replaced by its Fourier transform. The theoretical description has also had to be broadened with a movement away from periodic lattices to aperiodic ones. The lattice gas interpretation of the spin-^ System is given exphcitly äs follows. The eigenfunctions i/^^ of er 3 with eigen value + 1 correspond to states with a particle at x and the eigenfunctions i/^_ with eigenvalue -l represent States without a particle at x. The spin raising and lowering operators the
=
al^(a\^w\}l2, then act
äs
particle
a,
=
(a\-ia\]/2
,
creation and annihilation operators, e.g.,
:(cr^
+
1)=0
(a^+1)a,
=
((T^-1)a:
=
0-a,(ff^-1)
,
,
and hence
=
a^a^
=
-(ßx'A+),
measures
ffsK'/'-)
the number
''A
=
(zero
=
or
+K'/'-)
-
one) of particles
at
x
and
^X jceA
is the number operator for the Subsystem A. Now let Q be a simultaneous eigenfunction of each value -1. It follows that fljc^
10)
=
0 for all
jc
G
a\,x
^
A and the vectors
/^, with eigen
\X}
defined
by
Q and
i^)-n
form
an
orthonormal basis of
composition
9)^. Thus,
a
general
vector ^ G
A
^^s
a
de-
426
Models of
Quantum Statistical Mechanics r
v^
^
^
=
^(X)\X}
^CA
with
^(X)
eC and
E i^wi'
^
^CA
This construction is also
Z^' possible with A replaced by L and in the case L discrete form of the Fock spaces used to describe continuous Systems in Chapter 5, i.e., a vector ^ G /v is a function over the finite subsets X c Z' such that one
obtains
a
\\^f= ^ |T(X)P<+oo
.
XcZ''
Note that the creation and annihilation operators and anti-commutation, relations, e.g.,
[a,,<]=0,
satisfy
mixed commutation,
ifx^y
but
K,^*x}-i
.
Nevertheless, Example 6.2.14 shows that this model is in equivalent to a Fermi lattice gas.
a
certain
In the lattice gas interpretation the phenomenon of magnetization sponds to a form of liquid-gas phase Separation. The regions with the
sense
corre-
spins
up
correspond to regions occupied by particles (the liquid phase) and the complementary region with spins down corresponds to zero particles (the gas phase). The Heisenberg Hamiltonian has an interesting interpretation in the lattice Z and that the interaction is only gas language. For simplicity assume that L between nearest neighbors. The total Hamiltonian can be divided into several parts. First there is the l^-7contribution
^ J((T'{af^+CT-5(7fl)/2
H,=-
=
^
-
X
Therefore,
on
the
=
one-particle
(//,T)({j})
^
J(a'^a^+i +fl^fl*^J
space
-J(V^^)({y})
where V" is the discrete form of the
(V'-^)({y}) Thus, H\ is
=
.
n
-
2J^({y})
,
Laplacian
^({y + 1})
-
2^({y})
+
^({y
-
1})
.
analogous to the Hamiltonian of the ideal gas and the X-Ymodd is interpretable äs a noninteracting lattice gas. On the other hand the Ising contribution to the Hamiltonian,
Notes and Remarks
\
rjr
//2
=
^
_.JC+1 /'^ L
7-_x
2^ 7(73(73
-
l
427
,
:c= A/
has
pari 7/2
a
?
//^ which
=
V a>;+ifl;c+iöx
-2J
corresponds to the potential neighboring particles,
energy
arising
from
a
two-body interaction
between
(//^ ^)(7)
:^
(v
-/
Y.
^->MV^^)
^-j^ey
/
Thus, the Heisenberg model represents with
a
lattice Version of
a
particle System
interactions.
two-body
early work on spin Systems centered on spin-^ Systems with solely between neighboring atoms. The study of higher spin and intensified in the 1950s and 1960s. The general algeinteractions long-range braic formalism that we use was introduced by Robinson [Rob 9] in 1967 but it is basically an extension to quantum Systems of a framework given for classical Systems by Gallavotti, Miracle-Sole, and Ruelle [Gal 2] [Rue 1l]. The remark after Proposition 6.2.3 is due to Araki [Ära 13]. The first exact results on time evolution were derived by Streater [Str 2], for the Heisenberg model, and Robinson [Rob 7] for a general class of spin Sys tems. In particular, this latter reference contained a version of Theorem 6.2.4 in which it was assumed that O(J^) vanishes for \X\ sufficiently large. The ex tension to the case \^\^ < +00 was subsequently pointed out by Ruelle (see [[Rue 1]]). These results were not stated explicitly in terms of unbounded derivations but, in fact, motivated the development of integration theory for Most of the
interaction
derivations, and the consequent deduction of results such The first Statement in this theorem
the second If O is
by a
was
Bratteli and Kishimoto
established
can
be
easily
Theorem 6.2.6.
extended to O
^(X), O(7),...
commute and
satisfying
||0||o =sup5: ||0(^)||<+oo xeLXBx
because of the estimate
ii^"(^)ii
=
H
Y. ^inA7^0
^nA7^0
<2"|A|(||a)||c
[Kis 1] and
[Bra 1l].
classical interaction the various
Theorem 6.2.4
by
äs
Kishimoto
m^i},[---m^.,},m
428
Models of
Quantum Statistical Mechanics
for A e^\. Bratteli and Kishimoto
[Bra 1l] have given
which is "almost classical." If
< +00
||O||Q
a
generalization
to >
and
||[//c,(A),//ci>(AO]||<M for all
A, A' c Z, then the first criterion of Proposition 6.2.3 can be verified and dynamics is once again given by a strongly continuous one-para-
hence the
meter group
T^.
Another class of interactions
was considered by Matsui [Mat 4] [Mat 5] in 1993. We first discuss these interactions in the spin4 case, i.e., A^= 1. To understand the following formalism note that each element in the local
algebra
^A
can
be
uniquely expressed by
^
a sum
C^,rtT3W^i(n
X,YCA
where
Cxj are scalars, cr/(Jf) H. This is an easy conHxexf^] and (T/(0) seqiience of the Pauli spin relations of Example 6.2.14A, or the representation of the matrices in Example 4.2.7. Matsui proves that if the formal Hamiltonian is =
=
//=^T,(Ä) X
e
Z^'
where /z
=
Vc^,r
Y
and
^|C;,,r||^u7|-
then
a
of the with
resolvent estimate in terms of 'oscillation norms' shows that the closure corresponding derivation on y^ ^A is a generator. The
correspondence
our
interaction O is given by
Y^
^(X)=
Q,ßT,(
x,A,B
where the
sum
Thus
derives the estimate
one
is
over
all
z
x
G Z^'
and A,B c Z'' such that
||owil|x|
*'
=
(A ^x}^(B -^x)
=X.
^|Qß|Mu5|2
X3Q
For
general N U"^V",m,n 0, l,
one
=
U
.
.
/l
0
0
CD
.
,
can
expand the (A^
A^ where U and F
.
0
0 \
.
0
0
=
,
0
0
Vo
0
'. .
.
co^-^ 0
0
co^/
1)
+
are
x
(A^+ l ) -matrices
in powers
the matrices
v
/o
1
0
0
.
.
0
o\
.
.
0
0
.
.
0
1
.
.
0
o/
=
0
0
Vl
0
Notes and Remarks
and
co
expand
=
exp{27rzy(A/^+ 1)}
h in terms of
obtain similar results.
to
elements
positive
"^(X]
6
^x
äs
deduce that the closure of the associated derivation
Alternatively //
on
=
^^ ^(X]
the
one
429
can
and then
spin algebra
is
a
generator if
Y,\\^(x)\\\x\^<^ X
The connection to
formalism is
our
o(x)
=
now
Vi^m^-x)) E
(see [Mat 6], Theorem 2.9). The commutator estimates of Proposition 6.2.9 and
Corollary 6.2.10 are due application to the construction of the dynamics T^, Theorem 6.2.11, was subsequently given by Robinson [Rob 1l]. Despite the early development of the Ising model and the Heisenberg model the pure Jf-F model was not studied until 1950 when Nambu pSFam 1] gave a partial analysis. In 1961, Lieb, Schulz, and Mattis [Lie 2] established a preliminary form of the equivalence between the one-dimensional model and the Fermi lattice gas given in Example 6.2.14 and used this to calculate various facets of the model. The transformations used to establish this equivalence date back to the work of Jordan and Wigner [Jor 1] in 1928. The operator T äs well äs the other results on the dynamics of the X-Y model in Examples 6.2.14A and 6.2.14B are due to Araki in 1984 [Ära 3l]. Earlier work in this direction was done by Emch and Radin [Emc 2] and Abraham, Barouch, Gallavotti and Martin-Löf [Abra 1] [Abra 2] in 1971-72 and by Araki and Barouch in 1983 [Ära 32]. The return to equilibrium of the locally perturbed X-Y model is discussed by Hume and Robinson [Hum 1]. Throughout Section 6.2.1 we analyzed the time development of quantum spin Systems äs a norm limit of the local dynamics. There exist some long-range to Lieb and Robinson
[Lie 1}.
interactions for which this nevertheless
Their
norm
limit cannot be established, but where
prove that the
appropriate limits exist in certain representations and define automorphism groups of the weak closure of the spin algebra in these representations. Some such Systems were studied by Pulvirenti and Tirozzi in 1973 [Pul 1]. They considered a spin-j System on Z^'. In terms of the spin raising and lowering operators a* and a^ introduced earlier in the Notes and Remarks, the finite-volume Hamiltonians are deüned by one
can
//(A)
^Y.^(^- y^<^y + E J(X}a*(X}a(X] ^.>'eA x^y
jfCA
where
a*W=n.:,
a(X}
xex
Here K is
a
real function
on
Z^
satisfying
=
l[a,. xex
,
430
Models of
Quantum Statistical Mechanics
^|^W|<+oo x^Q
and / is
invariant real function
translationally
a
on
the finite subsets of /^'
satisfying
^\J(X)\<+oo
.
X3 0
One
can now
show that the local Gibbs states COA,
'"^^^^converge in the weak*
topology
activity exp{-jßJ(0)}
are
small
to
TrgJ.:^-!^^) Trg,(e-WA)) state
a
enough.
co
if the inverse temperature ß and the region of convergence one can
In the
derive estimates of the form
K[//(A),4^")*[/f(A),^]("^))|
^X+1)K'^+1)!^7"
<
for ^ G
UA^A^ where KA and CA are constants independent of A and the ^](-) are defined inductively by [//(A), A, [//(A),
[//(A), [//(A),
=
A]^^^
[H(\),A]^''~^^]
for
/2
-
1,2,
....
77, where ö is the closure of the derivation defined
from the interaction >. If //co is the
A]^""^
=
One next shows that^^ G
UA ^A
on
for all
D(ö")
in the usual way
Symmetrie operator defined
on
7rco(D((5))Qcü
by
iH^7i^(B)Q^
=
n^(ö(B}}Q^
it follows from the estimates above that
7ico(UA^A)^co
elements for //^o- Hence, the closure H^o of //^j is
^t(ncoW} by Proposition
-
3.2.58. In this
lim A>
COA(T;^(^I)
e^'^-n^(^fe-^^^-
case t
group associated with the pair after Corollary 6.3.30 that
-^
i^ßt is
(7i(^(^Y ,Q.a}}^
t^l^^O)
,
=
just
consists of
selfadjoint, =
n^(^y^
the modular
.
.
automorphism
and it follows from the Remark
^(^., (^i)
'
'
^JA))
oo
for all y4i, G [R. In the last expression ,^ G ^, ^1, , ^;^ its normal extension to na^C^Y and ^ with 7ico(^). .
analytic
and
.
.
.
we
identified
co
with
Section 6.2.2 The Gibbs ansatz for the
equilibrium
state of a
closed finite System
was
given
Willard Gibbs in the late nineteenth Century. Nevertheless, it is only since the mid-1960s that criteria such äs the Gibbs condition of Definition 6.2.16
by
have been
proposed äs characteristic for the equilibrium Subsystems of an infinite System. The first form proposed independently by Dobrushin [Dob 1], [Dob 2] the finite
of open Systems, of this definition
and
by
i.e., was
Lanford and
Notes and Remarks
Ruelle
[Lan 2].
These authors considered classical
form of Definition 6.2.16
proved
a
introduced
was
spin-^
by Araki and
431
Systems. The general
Ion
[Ära 24]
who also
version of Theorem 6.2.18.
The Dobrushin-Lanford-Ruelle version of the Gibbs condition is usually phrased in a measure theoretic manner which is quite different in appearance to the general definition. To illustrate this let us adopt the lattice gas notation given in the notes to Section 6.2.1. The classical C*-algebra GA of observables associated with A c L is taken to be the algebra generated by the identity and the number operators HX a'^a^. This algebra is *-isomorphic to the algebra of complex continuous functions over the finite subsets JT c A, e.g., the operator
/=E/wfn''.)fn(i-^) \yey
7CA
is
a
multiplication operator, f\X}
teraction 0 is then
/
\zA\7
f(X)\X)
=
for all X
c
A. A classical in-
operator of this type,
an
^\X)^^(X)\X) where the real function
^(X)
,
represents the energy of the
particles
at the
points
xex. Now the Gibbs state COA is determined
^(^}
=
by
its values
on
the monomials
Yl:ceX^^^^^ CD^(n(X})
=
V Z^ fi^(XuY)
,
rc A
rnA'=ß
where
g-Z.c^4'(S)
t^^(X) Note that ^^ is a probabiHty gebra is *-isomorphic to the
subsets ^ C
= -
2^^^,-I.,.0(.)
measure on
complex
the C* -algebra (i^A because this al
continuous functions
over
the finite
A, e.g.,
^^^(f)
E
=
/W/^AW
-
XC/i
Moreover COA is determined
by ^^, and vice versa. This is just an explicit representation theory of abehan C*-algebras given in Chapter 2. satisfy the equations
Version of the But the //^
MjJTuy)
=
e-^W-'(^l^);.^(7)
where
H(X)
=
Y^ 0(5) Scx
is the energy of the
particles
at the
points
of X and
,
432
Models of
Quantum Statistical Mechanics
W(X\Y}
^
=
5n
AV0.
^(S)
Sr^ J
^
ScX^
is the energy of interaction of these particles with particles at the points of Y. Thus, if one formally takes the limit A -^ oc, then {.i^ is replaced by a probability measure /,f on the C*-algebra ( generated by the UACI ^^ which satisfies the Dobrushin-Lanford-Ruelle equations
^(XuY) Each such
measure
=
/i determines
oj(A)
e-^^^^-^^^^\^^^i(Y} state
a
co over
(
^A(X}^i(X)
=
.
by .
XCL
It is
readily
checked that
satisfies the Gibbs condition in the
co
nition 6.2.16 and the Dobrushin-Lanford-Ruelle
sense
integral equations
of Defi
are
just
a
reformulation of the Gibbs condition. Both Dobrushin and Lanford-Ruelle
6.2.18 in the classical and Ion
[Ära 24]
spin-^
proved special
versions of Theorem
The first quantum Version was given by Araki for one-dimensional Systems and multidimensional Systems at case.
high temperature. The general version [Ära 25].
was
subsequently
established
by
Araki
Sections 6.2.3 and 6.2.4 The notions of entropy and conditional entropy have had fruitful in a variety of subjects, coding and information theory, ergodic
applications theory, sta-
tistical mechanics,
etc. This universahty is understandable because these quantities determine the asymptotic behavior of the multinomial coeificients and the multinomial distribution. Thus, they play a fundamental role in any application of basic statistics.
Let
denote the multinomial coefficients
"Cn^^...^n,
"
"c ^"^-"" where n\^n2^ one can
is
divide
large it is of with proportions n
objects
into
'
-l!. --,!
These coefficients express the number of ways subsets of /7i,2, ,m objects, respectively. If
h /^m
n
-
m
.
.
.
interest to examine the number of p\
=
n\/n,
.
.
.
Pn(p]
,
/>,
=
nt^/n, Thus,
partitions
one
into
m
subsets
examines
=
(p\n)\---(pmn)\
with p\ -^ P2^ l- But these coefficients h pm of the Stirling-type bounds =
^2^ /7'^e-''^i/(^-''+^)
<
nl<
are
easily estimated by
V2^n''e-''e^^^~''
use
Notes and Remarks
given by Robbins [Robb 1].
If R
and S
are
defined
433
by
R,:(p)
=
-(,-i)/2frf"' .^-l/2 Pi) (2nnr^"'-'^/M[l,
^(p^
=
-Y.p' '8
P'
i=l
finds
one
R..( r,\f>"S(P') R(p)e"^(p)
^
_
-],
0
=
n^oo
Pn(p] Alternatively
\n-^ logP,(;7)-^(;7)|=0(/7-Mog) 00. Thus, the predominant asymptotic feature of the P is an exponential increase, expfn^'}, where 5 is the entropy associated with the parti-
äs
>
77
tion p\,...,pn,. Next consider
n independent repetitions of a trial with m possible outcomes corresponding probabilities ^1,^25 5^w The probabihty that the zth event occurs with frequency /?/,/= 1,2, ...,m is given by the multinomial
and
distribution
Pn(p\q]="Cp,n
;,?r"'---e"
Therefore, the previous estimates allow
one
Rn(pY^^P\^^ -
l
to deduce that
=
o|i
Pn(p\q] äs
>
00, where m
S(p\q]
=
-
^(Pi log
Pi
-
A-
log qt)
i=\
is the conditional entropy of the frequencies {;?/} with respect to the probabihties {g/}. But S(p\q] <(^ with equality if, and only if, pi qt for =
i
l, 2,
tends to
.
.
.
,
7?2.
Thus,
zero äs 77"
äs /7
^'"~^^/^.
exponential decrease
S(p \ q). Therefore,
-^
00
each of the individual
But if the pi and ^/
are
exp{77*S'(;? |^)} governed by
not
probabihties pn(p \ q] equal, then there is an
the conditional
the
entropy
only results which effectively occur are those for which the frequencies closely approximate the probabilities. Note that this substantially reduces the effective number of results below the theoretical maximum of 77z'' exp{/7 log m}. The effective number is measured by the sum of the Pn(p] in a neighborhood of /?/ ^/. Thus, one obtains from the above =
estimates that
-
But if
one
log (effective number)
compares this with the
=
identity
iS'(^)
4-
oi -log
-
Models of
434
Quantum Statistical Mechanics
log(total number)
-
that the entropy
one sees
S(q} replaces log
=
m
log
m
,
äs a measure
of the number of
effective outcomes of each trial. These estimates
to a certain extent the
explain
quantitative origins of the qualitative features it is useful to examine the notion of uncertainty, or unpredictability. Each trial has a certain quality of unpredictability which varies with the probabilities. For example, coin tossing with two equally probable outcomes is much more predictable, or certain, than spinning a roulette wheel for which there are 36, or 37, possible results. This indicates that if one restricts one's attention to trials with equal probabiHties m~\ then the uncertainty F is a positive increasing function of q^ q^^^ m, i.e., entropy. To understand its
=
.
.
=
0
for m\ <m2- But each trial with m 1,^2 equally probable outcomes can be considered äs the composition of two independent trials with m{ and m2 out comes and it is natural to assume that the total uncertainty is the sum of the
uncertainties of the subtrials,
i.e.,
F(m\m2)
^
F(m\) ^F(m2)
But the
only functions which satisfy these integers mi and mi are given by
F(m) where
c
two
log
m
simple properties
for all
positive
,
a factor which determines the choice of units, the Thus, be to measured uncertainty appears by the logarithm of the number of possible outcomes. This is clearly not the case for trials with unequal probabilities. For
>
c
example, abilities
0.
if
are
up to
one
outcome has
probability dose to one, and all other prob no uncertainty. The result is predictable effective outcome. But in the preceding paragraphs
small, then there is almost
because there is
only
one
argued that
the entropy measures the effective number of outcomes. Thus, combination of these arguments justifies, to a certain extent, the interpretation
we
of the entropy
"missing
of
uncertainty. This interpretation appears to date observation, in 1894, that the entropy is related to
äs a measure
back to Boltzmann's
information" inasmuch
which remain
concerning expression
possible
to
a
i t is related to the number of alternatives
physical System
it has been recorded.
of maximum
äs
Thus, uncertainty,
after all the observable information
any
principle
or
minimal
of maximum entropy is an information, or maximal
disorder. As the entropy is of such basic Statistical significance it occurs in a wide variety of contexts and it is impossible to describe all such applications in these notes. It is the fundamental
great importance äs
an
in
concept in information theory and has been of
ergodic theory where a Version of mean entropy serves isomorphy invariant, the Kolmogorov-Sinai invariant, for dynamical
Notes and Remarks
Systems. Details of these and other applications Ornstein
Jacobs
can
435
be found in the books
[[Orn 1]],
by
Shannon and Weaver
[[Jac 1]], Billingsley [[Bil 1]], [[Sha 1]], and Yaglom and Yaglom [[Yag 1]]. The quantum-mechanical, or noncommutative, definition of entropy Tr(p log p) was introduced by von Neumann [[Neu 2]] in the early 1930s but there appeared to be no systematic investigation of its basic properties, e.g., subadditivity, concavity, quasi-convexity, until the mid-1960s. Nevertheless, most of these elementary properties appear scattered through the literature. The least known appears to be the quasi-convexity but a proof of this was given in 1947 by Jost [Jos 1]. A survey of these early developments has been given by Huber [Hub 1] who attributes the basic inequality of Lemma 6.2.21 to Delbrück and Moliere in 1936. appears to be the only significant established which of not was prior to the 1960s. In 1967 A.\-^S\(co) property Robinson and Ruelle [Rob 10] pointed out that this property could be derived
Strong subadditivity, Proposition 6.2.24,
spin Systems from the Kolmogorov-Sinai theory of the entropy of dynamical Systems. They also conjectured that it was valid in general. Despite attempts by many authors this conjecture remained unverified for 6 years until Lieb and Ruskai [Lie 3] gave a proof based upon a result of Lieb [Lie 4]. This
for classical
result States that the map Ai-^
-Tr(exp{L+^})
positive n x n matrices to the reals is convex for each selfadjoint L. partial results had been obtained by Baumann and Jost [Bau 1] and Araki and Lieb [Ära 26]. Another proof of strong subadditivity which uses somewhat more sophisticated notions is based on the monotonicity of the relative entropy under Schwarz mappings. This latter property was established by Uhlmann [Uhl 1]. A füll exposition of this second proof is given in the book of Ohya and Petz [[Ohy 1]] which also contains an exhaustive discussion of most features of the
from the
Prior to this
quantum entropy and related concepts. The systematic analysis of the entropy and
mean
entropy
äs
functionals
over
C*-algebras began with the papers of Robinson and Ruelle [Rob and Lanford and Robinson [Lan 5]. The first of these dealt with the abelian
the States of
10] algebras
of classical statistical mechanics and the second extended the form-
alism to quantum statistical mechanics. The main emphasis of both papers was on invariant states. Their aim was to prepare the ground for a discussion of the free energy, invariant of Section 6.2.4. The lavotti and
equilibrium states, and the maximum entropy principle thermodynamic free energy F was constructed by GalMiracle-Sole [Gal 2] for classical spin Systems and these authors
also described the connection between invariant limits of local Gibbs states and
tangent planes
to the
graph
of F. In
particular, they proved
a
version of
Theorem 6.2.40 and versions of Observations l and 2 at the end of Section 6.2.4. Ruelle
[Rue
1
1]
then used the entropy results and the results of Gallavotti (1) of Theorem 6.2.40. Subsequently,
and Miracle-Sole to prove property
436
Models of Quantum Statistical Mechanics
Robinson
proved The
[Rob 9] extended the whole discussion
to
quantum spin Systems and
Theorem 6.2.40 in the form stated.
identity
of the
mean
Gibbs States, mentioned after
entropy and the
mean
Proposition 6.2.38,
was
conditional entropy for by van Enter
estabhshed
[Ent 1]. The fact that they do not coincide on the invariant states follows from [Tem 1] (see, also [[Tem 1]], Chapter 8, Sections 5, 6 and 7). The next Step in this development occurred when Robinson [Rob 7] condynamics of a large class of quantum spin Systems äs a group of
structed the
*-automorphisms T^
and
proved
State, then this state satisfies the
result to all 0 for which T^ exists
Robinson
6.2 A3, the
ßHc^
is maximized
by
a
unique
condition. The extension of this
subsequently derived by Lanford and analysis. In particular these authors imphcation (3) =^ (1) in Theorem 6.2.42, Ob-
[Lan 6] using results of
proved Lemma
that if ^
(i^,ß)-KMS was
convex
servations l and 2 at the end of Section 6.2.4, and a Version of Observation 3 in which the weak* topology replaces the uniform topology. The weak*-version of Observation 3 was derived by proving that tangent functionals to a convex
continuous functional F
are
weak* dense in the F-bounded functionals. Several
years later in 1975 Israel
[Bis 2]
on convex
sets
[Isr 1] pointed out that the work of Bishop and Phelps implied that this result was true in the uniform topology.
Israel also
proved Observation 4 and various other interesting results concerning uniqueness and nonuniqueness of invariant equilibrium states. Theo rem 6.2.42 was finally completed by Araki [Ära 25] in 1974 with a proof that (2) =^ (3) and hence, by Corollary 6.2.19, that (1) =^ (3), i.e., the converse of the Lanford-Robinson result. An alternative discussion of lattice gases, and in particular the convexity arguments used in the last part of Section 6.2.4, is contained in the book by
Israel
[[Isr 1]]. Despite the
order of the text,
analysis of general, i.e., noninvariant, equili principles of maximum entropy did not make much headway the 1970s. The key Steps in this analysis were made by Araki and Sewell, independently and in collaboration. Araki [Ära 27], [Ära 28], [Ära 30]
brium states and until
both
introduced the relative entropy, Definition 6.2.29, and derived most of its basic properties. In particular, Propositions 6.2.32 and 6.2.33 and Corollary 6.2.34 are given in these papers. (Our definition of the conditional differs
entropy
slightly from that of Araki. One has S((p\il/) ^Araki(V^I^)-) An independent proof of the monotonicity property of Proposition 6.2.33 was subsequently given by Pusz and Woronowicz [Pus 2] by arguments based on their theory of the functional calculus of sesquilinear forms [Pus 3]. Kosaki [Kos 1] derived an interesting variational expression for the relative entropy, based on some of the ideas of [Pus 2], which can be used to estabhsh the monotonicity together with other basic properties. This line of reasoning is described in [[Ohy 1]], Chapter 5. The review of Wehrl [Weh 1] relates various properties of entropy and relative entropy. =
There remain many aspects of non-commutative entropy which we have not e.g., the attempts to extend the Kolmogorov-Sinai invariant from
described,
Notes and Remarks
437
dynamical Systems to von Neumann algebras. There are at dynamical entropy in the non-commutative of that St0rmer and Thirring [Con 5], that of Narnhofer, Connes, setting, Alicki and Fannes [Ali 1] and that of Voiculescu [Yoi 1] [Yoi 2] [Voi 3]. The book by Ohya and Petz [[Ohy 1]] contains more information on these topics and a good bibliography. Sewell [Sew 2], [Sew 3], introduced the principle of local thermodynamic stability for classical Systems, i.e., the principle of maximum conditional en tropy described in Proposition 6.2.35 and Theorem 6.2.36, and proved that it was equivalent to the Gibbs condition. Subsequently, Araki and Sewell [Ära 22] extended the notion of local thermodynamic stability to quantum lattice Systems, proved that KMS states satisfy this principle, and also proved that the converse is true for invariant states. Finally Sewell [Sew 1] derived the converse without invariance. The key to his proof is the autocorrelation lower bound discussed in Chapter 5. Attempts to analyze homogeneous Systems without invariance, e.g., impurity models, spin glasses, aperiodic Systems, from the present perspective can be found in [Rue 9] [Roo 1] [Ent 2] [Gee 1] and references cited therein.
measure
theoretic
least three different definitions of
Section 6.2.5 Both results of this section have
temperature
were
a
relatively long history.
basic type that we use for uniqueness at high first derived by Mayer and by Kirkwood and Salzburg for
Integral equations
of the
same
They are often referred Kirkwood-Salzburg equations. These equations, variants due to Mayer and Montroll, and various related developments of the 1945-1955 period are discussed in the book by Hill [[Hil 1]]. Originally the integral equations were used to study finite-volume equilibrium states. The first application of these techniques to infinite Systems occurred in a little known paper of Bogoliubov and Khatset [Bog 3] published in 1949. Basically these authors established the uniqueness of the thermodynamic limit for classical continuous Systems at high temperature and low density in the Gibbs canonical ensemble. Much later, in 1963, and quite independently, Ruelle [Rue 6] derived similar results in the grand canonical ensemble. Moreover, Ruelle [Rue 7] developed a technique which leads to spatial düster properties of the high- temperature states. This is an aspect that we have Z^', practically omitted from our general discussion. We mentioned that if L uniqueness of the KMS state co^ implies extremality and hence strong clustering of the form
correlation functions of classical statistical mechanics. to äs
=
lim
{o}^(AT,(B)}
-
(D^(A)co^(T,(B))}
=
0
I^HOO
for all A, B e^. But Ruelle's methods allow the deduction of
results,
e.g., if
^,5
G
91{o}
^nd <^ is
a
finite-range
more
interaction then
precise
438
Models of
Quantum Statistical Mechanics
|co^(^T,(5))
-
co^(^)a;^(T,(5))|
<
^"^^"l-^
for
some Co- (The details for quantum spin Systems were given by Greenberg [Gre 2].) Ruelle's work was generalized and extended by various authors, Ginibre, Gallavotti and Miracle-Sole, Greenberg, etc. Most of these general-
izations
are
discussed
Greenberg's after publication
work
in
Ruelle's
book
[[Rue
1]], the exception being
quantum spin Systems [Gre 1], [Gre 2] which occurred
on
of this book. The
popularity of these methods led Bogoliubov republish their work and it now exists in a more accessible English language version [Bog. 4]. In spin language the high-temperature-low-density regime corresponds to high temperature and weak external field. The extension of the uniqueness results to high temperatures and all external fields was given for classical spin Systems by Dobrushin [Dob 3] and by Gallavotti, Miracle-Sole, and Robinson [Gal 3]. The latter authors subsequently extended this result to a class of quantum spin Systems [Gal 4] using a method of Ginibre [Gin 1] but their work was quickly superseded by that of Greenberg [Gre 1]. The crucial fact that the Kirkwood-Salzburg type equations used by Greenberg could be derived from the KMS condition was first pointed out by Lanford [Lan 7]. Thus, Greenberg's results on uniqueness of the thermodynamic limit immediately translated into Statements of uniqueness of KMS States at high temperature. All these investigations of spin Systems were restricted to the simplest case, spin-j, and the present extension to general spin, Theorem 6.2.46, uses a slightly different technique. The first proof of uniqueness of equilibrium states for one-dimensional Systems was contained in the thesis of Ising [Isi 1]. The simplest method to obtain this result is by a method of Kramers and Wannier [Kra 1] and Montroll [Mon 1]. The basis of this method is the observation that the partition function of a chain of n spins with nearest-neighbor interaction J and and Khatset to extend and
external field /z, r
Zn(J,h}=
X! o-,=l
can
,.-1
X^ exp<^ -^Jö-/ö-/+i ö-
=
t
l
-/Z
i=i
be written in the form
Z,.(J,h)=Y: ^ e-"^/\M"-%e-^"r-, i
=
\
J=\
where M is the hermitian 2x2 matrix defined
^-
^M++
fe--^-'^
e-^
_
'
M__,
Thus, Zn is easily computed in M, e.g..
M+_\
by
M__; -\ terms of the
e^
e-^^^^
eigenvalues
and
eigenfunctions
of
Notes and Remarks
n~^ log Zn(J,h]
lim /2
>
=
Amax(^,/^)
439
,
OO
where
/Imaxl-^, h) is the largest eigenvalue of M. Similarly, one may compute unique Gibbs state of the model by adroit use of M. The matrix M is called
the the
transfer matrix. Ruelle has extended this method to dassical
spin Systems with long-range multispin interactions [Rue 8]. This generalization uses a number of new techniques, e.g., the Perron-Frobenius theorem, and partially relies upon earlier ideas of van Hove [Hov 2]. Subsequently Araki [Ära 15] gave a noncommutative extension applicable to quantum spin Systems with finite-range interactions and hence derived a special case of Theorem 6.2.47. The first proof of the general result was also by Araki [Ara29] and relied upon properties of the relative entropy. The proof we have indicated is due to Kishimoto [Kis 5]. It should be emphasized that uniqueness of the equilibrium state for onedimensional Systems is only valid for 'short-range' interactions. Dyson [Dys4] proved that the Ising model with a very long-range ferromagnetic interaction does exhibit a phase transition at low temperatures. Fröhlich and Spencer [Frö 9] Subsequently improved Dyson's conclusion by showing that a ferro magnetic interaction 7 (x) l/\x\ suffices for a spontaneous magnetization at and
=
low temperatures. Finally the transfer matrix method has been of great importance for the analysis of exactly soluble models, and in particular a version called the corner transfer matrix
[[Bax 1]]. The method has also been used to prove the 'equivalence' of certain two-dimensional dassical Systems and appropriate onedimensional quantum Systems and thereby obtain useful information about the dassical System. For example, Araki and Evans [Ära 35] analyze the Ising model by this method in the spirit of the current text.
Section 6.2.6 Much of the theoretical upon
In
understanding of phase ferromagnetic Ising model [Fei 1] proved the existence
of the
properties
1936 Peierls
for the two-dimensional
magnetization plingy < 0, at all sufficiently m
transitions has been based in two dimensions.
of
a
nonzero
spontaneous
model, with nearest-neighbor
low temperatures. The
proof of Theorem
cou-
6.2.48 is
based upon Peierls' argument. Five years later Kramers and Wannier
dimensional model has
unique 0 ^or ß unique ß^ such that m termined by the equation a
[Kra 1] proved that if the two(inverse) temperature ß^, i.e., if there is ß^ and m ^ 0 for jS > ß^, then ß^ is de-
critical
a
=
<
sinh27^, This result
was
obtained
=
l
.
(*)
by exploitation of a symmetry between high and low one to relate high- and low-temperature expansions
temperatures which allows
Models of
440
Quantum Statistical Mechanics
of the
partition function for the model. Relations of this type are now commonly called duality relations. Kramers and Wannier also developed the transfer matrix method, described in the Notes and Remarks to the previous section, for the one- and twodimensional Ising models and used it to calculate the partition function of the one-dimensional model. In 1943 Onsager [Ons 1] extended this method and calculated among other things, the partition function of a two-dimensional cylindrical array with vanishing external field. Onsager's results corroborated the earlier Undings of Peierls and Kramers and Wannier. For example, the energy per atom of the infinite two-dimensional System is a continuous function of the temperature T ß~^ and is differentiable except at the Kramers-
mean
=
Wannier critical value TC. At this value the derivative
Subsequently,
in
diverges
äs
log \T
TC
1948, Onsager announced the value
m=[l
-
cosech^ 2ßj] 1/8
for the spontaneous magnetization m below TC but he never published a proof of this claim (see [Ons 2]). A proof was given by Yang [Yan 1] 4 years
later. In 1952 Lee and
Yang [Lee Yl] devised an alternative method of partially analyzing ferromagnetic Ising model, in an arbitrary number of dimensions. Their method consists of isolating the zeros of the partition function ZA for the finite-dimensional System. (These zeros potentially lead to singularities of the free energy FA |A|~MogZA in the thermodynamic limit and hence herald transition phenomena.) In particular, they discovered the remarkable the
=
fact that the
zeros
h is the external
of ZA, viewed
field,
lie
on
äs a
function of the parameter Qxp{ßh} where They could conclude that the
the unit circle.
free energy is differentiable in h except at /z 0. It appears that the success of the Onsager-Yang calculations for the twodimensional Ising model diverted attention from the general analytic techni-
thermodynamic
=
ques of Peierls and Kramers and Wannier and these methods
were
not
re-
analyzed until the post-1964 period. In 1964 the Peierls argument was revived and reformulated by both Dobrushin [Dob 4] and Griffiths [Gri 1]. Grifiiths claimed the original Peierls argument was incomplete but this was disputed by Peierls [Pei 2]. An early discussion of Peierls' argument was given by Wannier [[Wan 1]]. After the work of Dobrushin and Griffiths many extensions and generalizations of the Peierls method subsequently appeared. (For reviews up to 1972 see [Gri 2], [GriS].) In particular Dobrushin [Dob 4] showed how the argument could be applied to anti-ferromagnetic Systems, Theorem 6.2.49. Moreover, in 1968, Robinson [Rob 12] realized how the arguments for the classical Ising model could be extended by perturbation theory to the quantum-mechanical anisotropic model. The proof of the weak version of Theorem 6.2.48 is based on Robinson's original argument. Subsequently, Ginibre [Gin 2] reformulated various quantum models äs perturbations of classical models by the path method sketched at the end of the proof of Theorem 6.2.48 and thereby improved and extended Robinson's conclusions. Kennedy [Ken 1]
Notes and Remarks
finally optimized Ginibre's arguments
for the
anisotropic Heisenberg
441
model
and established the füll Statement of Theorem 6.2.48.
Hohenberg [Höh 1] was the first to realize that Bogoliubov's inequality [Bog 2] could be used to rule out critical phenomena. In particular, he proved the absence of Bose-Einstein condensation for
one-
and two-dimensional
continuous Systems. Theorem 6.2.51, the lattice analogue of this result, was subsequently given by Mermin and Wagner [Mer 1] but the absence of a
spontaneous magnetization for the Isotropie Heisenberg model in mensions
already folklore.
two di-
A
heuristic, and apochryphal, argument which Supports this result and also indicates the importance of the isotropy, in at least two directions, is discussed by Griffiths in [Gri3]. The origin of the Bogoliubov inequality given in Lemma 6.2.50 has been was
described in the Notes and Remarks to Section 5.3. The absence of continuous symmetry breaking described in Theorem 6.2.50 established by Fröhhch and Pfister in 1981 [Frö 6] [Pro 7] although their
was
method
local
was
based
on
uniform estimates
on
the relative entropy of suitable
of the KMS states, in the spirit of the proof of Theorem 6.2.47. But to obtain these estimates they essentially had to estimate ||^$(^L)||
perturbations
äs in the proof we have given. Our proof is based on the 1984 result of Fannes, Vanheuverzwijn and Verbeure [Fan 4]. It should be emphasized that the ab sence of spontaneous breaking of symmetries in one-, and two-, dimensional models with short ränge interactions does not exclude the existence of phase transitions in such Systems. Examples of such transitions in the classical case have been given by Dyson [Dys 3] and by Shlosman [Shl 1]. Note also that the estimate on the ränge on ||({^, 3^})|| in Theorem 6.2.50 is the best possible in
the
sense
that there exist two-dimensional Systems with
a
continuous symmetry y\~'^~^^, where
group G and two-body interactions ^ with ||({jc, j^})|| C\x > 0 is arbitrarily small, for which spontaneous symmetry breaking ~
can occur
[Kun 1] [Frö 8] [Bon 1] [Ito, K.l]. Theorem 5.3.33A
can
symmetry breaking for
also be used to estabHsh the absence of discrete
some
one-dimensional models with interaction of
ränge that Theorem 6.2.47 does not apply [Fan 4]. The method of infrared bounds was introduced by Fröhlich,
so
long
Simon, and
Spencer [Frö 1] who successfully applied it to a number of classical models, including the multidimensional Ising model. The crucial bounds are obtained by a method of Gaussian domination previously devised by Glimm, Jaife, and Spencer [Gü 8] to handle phase transitions in constructive field theory. If /z is a real functional over Z^ (dyh)(x) h(x -\- y) h(x') and =
,
=
Y,^lh(x)
,
X
then the Gaussian domination bounds for the
coA
(V (v ( exp
^3
E
\y;n.n.
^^-M
ll
///
<
Ising
model
are
expj^ (2^y)~' E I^Wl' Jl ^
442
Models of
Quantum Statistical Mechanics
where COA is the local Gibbs state and the y sum is over nearest neighbors. The derivation of these bounds uses the Kramers-Wannier transfer matrix method. It follows from them that
a>^(a(h)a(-^h))
(ßjT^ Y.\^(x)\^
<
'
X
where
-A/z(jc)
=
2vh(x)
Y,
-
j^^;n.
and this
model
can
be used to bound
displays
a nonzero
coA(ö-3Ö-'3).
2(2ßj]>(27Lr
/
7u-,-i<7r /|A-,-|<7r
gives
a
bound
correct value for
on
+
>^)
+
^(^
3^))
-
In
particular, one finds that spontaneous magnetization whenever r
This
(^(^
n.
/
the
Ising
^
^'
^^'^^(1-cos^,) \7:rf
the critical temperature which is within 10% of the
the two-dimensional model.
Apphcations of the infrared bound technique to quantum spin Systems have given by Dyson, Lieb, and Simon [Dys 2], and Fröhlich and Lieb [Frö 2]. These developments have been reviewed by Lieb [Lieb 5]. In particular, these been
authors derive the results
on
the X-Y model mentioned
near
the end of the
section and similar results for the
anti-ferromagnetic Heisenberg model. isotropic Heisenberg model and resistance in electrical networks was pointed out by Powers in 1976 [Pow 6] [Pow 7] [Pow 8]. He established a connection between the existence of long ränge Order and the resistance properties of the graph of an associated electrical An
interesting
connection between the
network. The
multiplicity properties of equilibrium states of the /"-invariant twoferromagnetic Ising model follow from the work of Ruelle, Messager and Miracle-Sole, Aizenman and Higuchi [Aiz 2] [Hig 1]. Ruelle [Rue 9] proved uniqueness for nonzero external field, or for small ß, and Messager and Miracle-Sole proved that there are only two /--invariant extremal (T,^)KMS States in the regime for which the spontaneous magnetization is nonzero. In fact, Ruelle proved there is a unique state satisfying the Dobrushin-Lanford-Ruelle equations described in the Notes and Remarks to Section 6.2.2. But these equations are equivalent to the Gibbs condition for classical interactions and hence one deduces uniqueness for the KMS states. Messager and Miracle-Sole [Mes 1] proved that the tangent functionals to the free energy are uniquely determined on the classical interactions defined by even polynomials
dimensional
of the (73, e.g.,
on
interactions 0 of the form
)({^i,...,x2,})-(r5'...(r5^",
and
a)({^i,
.
.
.
,z2,+i})
-
0
.
It follows from the results of Section 6.2.3 that all the /--invariant
KMS states coincide
on
(T, ^ö)subalgebra generated by even poly follows from a simple argument which
the classical
nomials of the 0-3. The desired result
Notes and Remarks
443
Z^-invariant (T, jß)-KMS states form a simplex. An al given by Lebowitz [Leb 1]. Aizenman [Aiz 2] and Higuchi [Hig 1] established independently of each other that all (T, ß)-KMS states of the Z^-invariant, two-dimensional Ising ferromagnet are Z^-invariant. Messager and Miracle-Sole's results exploit the duality techniques introduced by Kramers and Wannier. At very low temperatures more detailed Information has been obtained on the phase structure of general classical models by combining duality arguments and the Lee-Yang method of zeros. A comprehensive review of these developments is contained in the monograph of Gruber, Hintermann, and Merlini [[Gru 1]]. There is one important general technique in the theory of phase transitions for classical ferromagnets which we have not touched upon in the text. This is the method of correlation inequalities. This method originated with GriiRths observation [Gri 4] that a ferromagnetic spin System with a positive external field should have a positive magnetization in equilibrium and this magnetization should increase if the interparticle coupling, or the external field, is increased. Quantitatively these observations are expressed by the inequalities uses
the fact that the
ternative derivation of this result has been
COA(^5)>0
^A(^3^3)
-
,
o}^(ff'l)o}^(a^}
>
0
,
for the local Gibbs
s täte COA. Griffiths' results were quickly generahzed by Kelly [Kell 1] who established the foregoing inequalities remain valid if 0^, 0-3 are replaced by products
and Sherman even
(^x=
n ^3.
Ginibre has
given
n
0^7=
xex
^3
yeY
proof
of these basic
inequalities [Gin 3]. Subsegeneralizations, e.g., Fortuin, Ginibre and Kasteleyn [For 1], Lebowitz [Leb 2], [Leb 3], etc. These inequal ities have had numerous applications, e.g., to prove the existence of the thermodynamic limits of Gibbs states [Gri 4], to establish uniqueness of equilibrium states [Rue 8], to deduce the existence of phase transitions in one-dimensional Systems with long-range interaction [Dys 3], to obtain bounds on critical temperatures [Gri 5], to establish rigorous inequalities on parameters determining critical behavior [Bück 1], and to bound high-order correlation functions by low-order correlation functions [Leb 3]. As an explicit example of this last type of application we remark that Lebowitz has proved that the Gibbs State COA of an Ising ferromagnet satisfies quently
an
easy
many authors derived
o^\(nxny)
-
COAK)<^A(%)
extensions,
^
or
5Z X^ o)^(n^ny)
~
o^h.(n^}^^(ny]
,
jcejr^er
where n^
=
n.^
is
the
ala^-=^ (ö-^
+
number
1])/2
and
operator in the lattice gas interpretation, i.e.,
444
Models of
Quantum Statistical Mechanics
;^
JJ
=
T
xex
Unfortunately, the Griffiths type of correlation inequality has only been derived for classical models and little information is known for genuine quantum Systems (see Ginibre's review [Gin 4]). There are a variety of other general methods which have been used to elucidate the behavior of models of quantum Statistical mechanics, e.g., mean field theory and quantum fluctuation theory. The thermostatic part of mean
field
theory is described in [Rag 1] and the dynamical part in [Duf 1]. The theory of normal fluctuations can be found in [God 1] [God 2] and the theory of abnormal or critical fluctuations in [Bri 1].
Section 6.2.7 This section is based upon the paper of BratteH, Kishimoto, and Robinson [Bra 15]. But the implication (2) =^ (1) of Theorem 6.2.58 had already been
given by
Ruelle
[Rue 10].
The condition
zenmann, Davies and Lieb [Aiz 3] [Aiz The conditions that T is positive and
imply that r(^A) pletely positive.
C
^A, T(AB}
The detailed structure of the 6.2.56
=
H
derived
=
V(l /
in the theorem is due to Ai-
see
also the
T(A) T(A)B for ^
ground
Example by [Mat 1]. The return to equilibrium in 1 984 by Araki [Ära 3l]. The ground states of the XXZ was
4],
(3)
appendix in [Köm 1]. ^A'^ automatically ^A, ^ e ^A^^ and T is com-
=A for ^ G G
states of the X-Y model described in
Araki and Matsui in 1985
-
^
[Ära 33] [Ära 34]
for finite temperature states
was
established
model with Hamiltonian
(T] crf
^ -
ö((r[ ^{+1
+
trj cr^^)}
.xez
where 0
^
l have been studied
by
Matsui
[Mat 2] and by Gottstein and ground states and down, respectively, there are two families of non-translationally invariant ground states called kinks which interpolate between the spin up and spin down states. The extremal kinks in each of the families are all unitarily equivalent. The detailed justification for the classical analysis of the ground states of the Ising model given in Example 6.2.56, and the connection between the condition given in Theorem 6.2.52 and the classical condition is äs follows. Assume ^{-c} Md for each jc G L and hence ^ (8);c6iM/- Let Q :^ C"^ be the subalgebra of diagonal matrices in M^ and ( (8),TiQ ^(^^^L^d) the corre sponding maximal abelian subalgebra in ^, the classical spin algebra. The <
<
Werner
[Got 1]. In addition corresponding to the spins up
~
to the two translation invariant
~
~
interaction O is defined to be classical if O
=
^(X)
X^ei^i/ is called the classical configuration
in Cd there is
an
G
x^Cd for each X ^L and Ud is the unitary group
space. If
obvious action of the compact
product
group
X^^i^d
on
^
Notes and Remarks
445
whose fixed
point subalgebra is exactly ^ and this is clearly a symmetry group for the classical interactions. This symmetry is never broken for finite temperature equilibrium states, i.e., the KMS states of the quantum model are diagonal states, striction to G
equations
are
and the
probability measures equilibrium states in the
on
Q
one
obtains from the
re-
classical sense, i.e., the DLR fulfilled. This follows from the Gibbs condition of Theorem are
6.2.18. On the other hand for
ground states this symmetry may be broken. This already in the matrix case. If, for example, the Hamiltonian has a doubly degenerate lowest eigenvalue the set of ground states is affinely isomorphic to the 3-ball by Example 4.2.7, but there are only two extremal classical ground states. Nevertheless it can be shown, and it is evident in the matrix case, that a state co on 21 is a ground state in the sense of Theorem 6.2.52 if, and only if, co|ß- is a ground state in the classical sense. This means that 0), viewed äs a measure on Q, is supported on the set of ground state configurations, i.e., the set of configurations that has minimal energy with respect to changing the configuration at a finite number of lattice sites. Another characterization is in terms of the projection g^ ^&A ^^ onto those local configurations in Z^ which have extensions to global ground state configura l for all finite tions; a state co G "21 is a ground state if and only if o}(gA) subsets A C L. Thus the set of ground states is the closed face in E<$i generated by the classical ground states, where a ground state is called classical if it is invariant under the action of the symmetry group Xx^L^d on 91, and thus it is uniquely determined by its restriction to the classical subalgebra. Nevertheless, one may have pure quantum ground states which are not unitarily equivalent to any ground state coming from a classical configuration. A simple example of this is given by the trivial interaction, but there may be more interesting ones like the Ising anti-ferromagnet on a triangulär lattice [Got 2]. Analysis of the invariant ground states of the one-dimensional Heisenberg model was initiated by Bethe [Bet 1] in 1931. The most extensive rigorous results have been derived by Thomas and Babbitt and Thomas [Tho 1], [Bab 1], [Bab 2]. They describe the spectral properties ofHa^ in great detail. It should be emphasized that our brief discussion in terms of the Separation of the Z- 7 and Ising contributions is meant to be purely illustrative. Other descriptions of the model äs a lattice Fermi gas (see Example 6.2.14) with two-body interaction or a lattice Böse gas with an additional hard-core interaction (see, for example, [Gin 1]) have also been used extensively. The X-Ymodoi, or the Ising model, in Example 6.2.56 show that there may be a great multiplicity of ground states, even translationally invariant ground states, in situations for which there is a unique (T, jS)-KMS state for each finite ß. Thus one cannot expect to find many general situations where there is a unique ground state. Matsui [Mat 3] proved for a class of translationally in phenomenon
can
be illustrated
=
variant interactions with such that there is ^ >
an
external field
^"^xez^^
unique translationally
invariant
^^^^ there exists
a
Ao
ground state provided AO- The class of interactions include the Heisenberg model with formal
Hamiltonian
a
Models of Quantum Statistical Mechanics
446
//
=
-A
^ X
a\
^
+
e Z'
|x->;l
where ai
<
0 and 02^3 < 0. The
invariant
(a\(T\(Tl
ci2cr\(J2
+
=
l
proof
uses a
representation
finite temperature states of ground Kirkwood and Thomas [Kir 1]. developed by
Finally
states
we
äs
comment
a-^(T]^(j^^]
+
.X, y E Z'
method of
on a
of
translationally spin System
classical
a
constructing translationally
invariant
one-dimensional quantum lattice Systems which has been used to the ground states of certain interactions. Let ^ (8)^^M^-|_i be the
pure states
on
analyze C*-algebra
of
=
a
quantum lattice System V
be
Z, fix
over
Define
a
completely positive
a
state on
=
/: G f\l and let
C^'h^C^'(8)C^+^
:
VAV. Let p be compatibility condition
isometry.
an
E(y4)
a
Mj^ and
map E
assume
:
M/,- (8)MAr+it-^M/: by
that
(p, V)
A)
for A G
satisfies the
p(^(BI)]^p(B] for all B eMk. Define EA one can
define
a
60(^1
:
Mk
0
.42 (8)
...
States constructed in this
^
M/, by E^ (B}
^
translationally
=
invariant state
A,n)
=
manner
P(EX, are
O
E(B
co on
E^,,^_,
called
o
^
^
by o
.
.
.
finitely
the
MA^+I Then requirement
E^j (4))
.
.
correlated
states
by
Fannes, Nachtergaele and Werner [Fan 5] [Fan 6]. In the second reference it is established that if are
co
is
a
finitely correlated
state then the
following
conditions
equivalent:
(1) (2)
co
is pure,
proper subalgebra of Mk containing Ik is invariant under E^,^ GM;v+i, E/ has trivial peripheral spectrum, i.e., the only eigenvector of no
(3)
E/ with
Using
eigenvalue
of modulus
all
:5h->r(5(g)/)F one
this construction Fannes,
is Ik.
Nachtergaele and Werner [Fan 7] have es translationally invariant pure states is weakly* dense in the set of all translationally invariant states. Thus the conclusion in Example 4.3.26 that the extremal translationally invariant states is weakly* dense in the set of all translationally invariant states on a quantum spin System can be considerably strengthened in the one-dimensional case. Results on phase diagrams for genuine quantum statistical mechanical models are relatively sparse compared to the results for classical models [[Sim 2]] [[Sin 1]]. Some results have been obtained äs quantum perturbations of classical models on the diagonal algebra. These results rely on the PirogovSinai theory [[Sin 1]] of classical ground states to obtain the complete phase diagram of the quantum perturbation at small temperatures [Bor 1] [Dat 1]. tablished that the set of all
Notes and Remarks
The method is based
on
the
assumption
that the classical System has
447
finite
a
number of pure classical ground states which are then perforce periodic. The models assume that the quantum Hamiltonian has the form H H^^^> -i- dV where H^^^ is (formally) in the diagonal algebra and V is the quantum per-
turbation. The conclusions
are
2] [Ken 1] described in Section e
1^" ^
drawn
by
a
6.2.5. One
Variation of Ginibre's method
[Gin
the Trotter formula
uses
limfe-/^^'"'/(1-^F/))"
=
?2
^
00
\
/
expand the perturbed density matrix e ^^ around the unperturbed one g-^//(o) ^^.^ expression introduces transitions between classical contours at various times and leads to a representation in terms of 'quantum contours' on [0,^] X f. In the case considered in [Gin 2] [Ken 1] the quantum contours could be controlled by Standard arguments whilst in the more general Situation encountered in [Dat 1] a quantum version the Pirogov-Sinai theory is required. In [Bor 1] one does not, however, consider these quantum contours but uses a to
map of the v-dimensional quantum System into
a
(v
-f
l)-dimensional
lattice, where the classical Pirogov-Sinai theory is applicable. This somewhat analogous to the maps used in [Ära 35] and [Kle 1].
block
map is
Section 6.3 The earliest
Systems
rigorous
were
dealt with
results
obtained
properties
by
on
the statistical mechanics of interacting quantum [Rue 12] and Fisher [Fis 1]. Both these papers
Ruelle
of the
thermodynamic functions, i.e.,
the pressure, free
energy, etc., and the first results concerning the equilibrium states of such Systems were given by Fisher [Fis 2] and Ginibre [Gin 5]. In particular, Fisher
emphasized the importance of convexity, and differentiability, of the thermo dynamic functions in terms of the interaction potentials for the discussion of phase-separation properties. These ideas were subsequently developed in the context of spin Systems into the theory of tangent functionals and invariant Gibbs states described in Section 6.2.4. On the other hand, Ginibre introduced the techniques of functional integration. He showed that the quantum-mechanical reduced density matrices could be represented äs integrals of func tionals which resembled the correlation functionals of classical Systems and exploited methods of classical statistical mechanics to obtain a variety of high-
temperature, low-activity results. These techniques, which are described in Sections 6.3.2 and 6.3.3, have subsequently played a fundamental role both in quantum statistical mechanics and in constructive field theory. In fact, Statement in the
introduction,
our
the effect that the ideal Böse gas is the only model of continuous quantum Systems in which phase transitions have been rigorously established, is a slight exaggeration. In constructive field theory
phase transitions have
been
to
rigorously
derived for the Euclidean
ories. To define these theories consider the CCR
particle
space
1^
consists of the distributions g
^
algebra ^+(t)),
^'(U~]
^(0)2
the-
where the
one-
such that
Models of
448
Quantum Statistical Mechanics
j d^-p\g(p]\\p^
=
Here g is the Fourier transform of g and completion of L-(R-) in the norm
\\g\\^^(cj^(-V^Let
I),.
that g cuum
denote the =
g, where
subspace
g((p]
of
m
+
m^Y^
is
for cp e
<+<^
positive constant, i.e., l)
a
real
distribiitions, i.e., of
and let
is the
^(U~)
CD
0^ such
denote the Fock
va-
State, i.e..
o4W(g))=&xp[-\\g\\^/4} where W is the
Weyl operator.
and
verifies from the
that
.
m^Y'9]"-
I) consisting of
g(cp]
=
+
one
easily
{W(f)]f
e
l)f.}
generates
Note that
W(f)
W(g)
and
decomposition 1^
a
,
=
maximal abelian
^,.
von
commute if
/,
l),,
^ G
and Lemma 4.3.15
il)^.
-|-
Neumann
algebra
9J1
on
Fock space. Let >(/) be the associated field operators (defined in Proposition 5.2.3.) The restriction of > to f),. is called the free Euchdean Markov field of mass m
in dimension 2. Let 9JI
let //o be the regulär Borel is called the free measure. To introduce jLiQ. Define
an
=
C(g)
be the Gelfand
interacting
defined
Q
measure on
Euclidean field
by
theories,
'^
where /i (V" -^ nr) of yi*(/?) and its adjoint =
>
jj.(p)
=
(p~
=
+
and
A(p]
state. /iß
the
measure
perturbs by
,
is the inverse Fourier transform of /. In terms
the Euclidean field is
j d^-pe~^'"^[A'(p]
(In)-^
m^)^^^.
/
of 9Jl, and
vacuum
one
operator- valued distribution ^* from a*
j d^-pA^(p)f(p]=a\^if)
where
representation
the Fock
/
Note that if
+
given by
the
A(- p]]n(p)-^
and g
are
expression
,
real, A and ^* satisfy the
commutation relations
[A(f],A^(g]]=^ One defines Wick powers of the field
J
7
.
by
/l
^"C,-^*(;7i) =
d^~xf(x)g(x]
/n(At(A-)-'^^,Oe-"''"+"'+''"'
:
X
j
-
.
A^(pj)A(-pj^,)
.
.
^A(-pn]
0
quadratic form on finite-particle vectors with smooth components. This corresponds to placing all creation operators to the left and all annihilation operators to the right in the expression for the "operator power" ^(x)'\ If P is a polynomial of the form as a
Notes and Remarks
449
2m
P(x}
=
^a.x^ r
with a2m
>
0 and fif is
a
sufficiently
U(g)
=
nice
=
0
function,
one
defines
j d^xg(x):P(^(x)): 2m
E']ajd^xg(x):'-(x}: J r
Formally
one
defines the
=
interacting
M/) but U (l) does not make finite-volume Gibbs
.
0)
Gibbs
measure
by
/to(e-''^"V) =
/Zo(e-/'^(")
sense äs a
function
on
ß, and
one
has to introduce
measures
/^O...A(^-^^'^->/) /^o..,AM^('^0
...
^^'"^^^
where A^o,(7,A is the measure defined äs //Q with V^ A in the with classical conditions placian boundary
replaced by r[^^, the Lagiven by a (see Example
5.2.26) and x\ is the characteristic function of A. One may also use //OJ.A /^o? i.e., the so-called free-boundary conditions, or a measure A^O,/?,A corresponding to periodic boundary conditions in this definition. It was estabhshed by Guerra, Rosen, and Simon [Guer 1], [Guer 2] that the infinite-volume pressure Hm A-.R2
exists and is
independent
-^log/.o,.,A(^'''^'^^) |A|
of the
boundary
Phase transitions in these models
were
conditions.
first exhibited
Spencer [Gli 6], [GH 7]. Specifically, they considered
P,^,(x)
=
^^'~f^\hx
the
by Glimm, Jaffe, and polynomial
.
Then, with Dirichlet boundary conditions, the Hmits of the
momenta
of the
fields
M exist for
,^,., ^,.,, W,)-(/))
/l, ...,/
G
^r(ö^^).
,.
f^,,^Ae-"^'^^^<^(fi)---^(fn)}
=^lim^,
,^^^,(e-.(..))
However, if
A: is
sufficiently large
one can
show
that lim n
for
positive /, i.e.,
the symmetry C[)
>
for h
^^
U +
lim
co,,,((D(/)) ^ h
>
co,,,(a)(/))
0
0 one has pure phases which are not invariant under
450
Models of
Quantum Statistical Mechanics
Peierls argument of Section 6.2.6. The local nature of the interaction makes these models very similar to lattice models, and one can use all the machinery from the latter models, i.e., correlation the former. A review of the
inequalities, etc., to prove results aboiit resiilting developments can be found in [Frö 3]. The
greatest physical interest of these models but from the fact that
they closely dimension (see [Giier 1]).
space
not from statistical
comes
mechanics,
related to quantum field models in
are
one
Section 6.3.1 It shoiild be
that the time evolutions
emphasized
Tf(^)z=e''^^-A^^-/^//.,A generated by
the local Hamiltonians
Ha, A
T'fT, A
=
+
?^A
be
expected to leave the C*-algebras ^(A) invariant if ^ ^ 0. A was proved by Fannes and Verbeure [Fan 3]. rigorous They consider the CCR algebra ^l+(f)) over f) C, i.e., the C*-algebra of ordinary quantum mechanics in one dimension. The Schrödinger representation, Example 5.2.16, realizes this algebra on L-([R) äs never can
result in this direction
=
(r(^)iA)W
=
e'--iAW
,
(W(is)ilj)(x)=ilj(x-s] for
G [R. Next
^
define 1
dW(is]
i
äs
P
=
J
dW(s)
i
ds
-
and
a
.=0
l
ß-
1^
.^0^'
Hamiltonian
HX=P~^^ V(Q] where K is
a^
of
d^
'
a
real, bounded function and A
,
G [R. Define
a
one-parameter group
^(L-(R)) by
üL](A)=e'^^-^Ae-^^'^-'
.
One verifies
a?(FF())
=
W(is')
,
oi%W(s))=e'-''''W(s(l-2it)) i.e.,
?i-^
is not
a deflnes
strongly
,
a one-parameter group of *-automorphisms of 2I+(C), which continuotis becatise of Theorem 5.2.8 (1). Fannes and Verbeure
Notes and Remarks
prove, however, that if A
(i.e., F(ö) e ^l+(C)\C1]) 51+ (C) into itself for ^ 7^
7^
0 and V is either nonconstant and almost
or
F G In
L^ and
F
then
/ 0,
a;-
451
periodic
does not map
0.
automorphism groups do not leave the local C*does not lead to any problems since the local Gibbs invariant algebras globally Fock the in normal States coA are representation, and therefore have unique The fact that the local
^(A)'^
normal extensions to
==
^(g.^(A)). Expressions
hke
fflA(^i)...T;^(^,)) for AI,
.
.
.
^Afi
M(A)
G
should therefore be
interpreted
äs
-Ane-^^"^^)
Tr^^(A)(e-^^Ag'-/,//A^j^/(/2-/,)//A^2 Trj^v(A)M^'0 etc.
The hterature on pertiirbations of the infinite- volume Laplacian -V^ by multiphcative operators U is extensive (see, for example, [[Kat 1]], [[Ree 2]]. [[Sim 1]]). There does not, however, appear to be any readily available description of the corresponding problem for finite-volume Laplacians with classical boundary conditions although several results can be easily transcribed from the infinite-volume case. For example, the equality of T^ f^ + [/^"^ and ^!"A + ^^"^ for U^"^ G L^(^T and U^"^ > 0 follows from an inequahty of Kato by the same argument used for V" (see Section X.4 in [[Ree 2]] for a description of this technique). Similarly Proposition 6.3.1 is a Version of Kato's theorem of 1951 [Kat 2]. The estimates of the kernel of the semigroup used in the proof are discussed in detail in Section 6.3.2. Stable interactions are discussed in detail in [[Rue 1]], and the Statements in Examples 6.3.2 and 6.3.3 are proved there. Example 6.3.2 is originally due to Dobrushin [Dob 5] and 6.3.3 was proved by Ruelle in [Rue 13]. In fact, he proved that if the condition
Y^^<^(x,-xj)>() is not
satisfied, then
region
A with
n,
where C
>
find
one can
configurations
of
n
particles
in
a
bounded
energy less than or equal to Cn- for infinitely many 0, which establishes catastrophic behavior of the classical System. a
potential
Furthermore, it is possible
configurations
cp^^ G
^"(A)'^
approximate
to
these
configurations by quantum
such that
(cp,,,U^cp)<-Cn^ and
r C?i
for bosons
,
,
^
(^;p^oo,A^J
< <
l C'07^+-/^'
for fermions
452
for
Models of
infinitely
Quantum Statistical Mechanics
many
n.
This shows that if the classical
stability condition
^A>-^A^A is
violated
for
an
semi-continuous
upper
two-body interaction
then
the
quantum stability condition
H^^^>-BN^ is also lead
always
one
violated for bosons, and, if
v
to believe that the two notions of
3, also for fermions. This could
>
stability
are equivalent, but in the important physical case of the Coulomb interaction of different species of particles they are not. This was pointed out by Dyson and Lenard in 1967-1968 [Dys 4]. Consider, for simplicity, n particles in IR^ with the mass \ and charges it 1,7 77. The Hamiltonian of the l, ey System is given by =
==
.
.
.
,
"n-t.-^]+ f
-^
E
'
J
7=1
f
J
Y'
\
The
particles may be fermions and/or bosons, general it is possible to prove the lower bound Hn
where B is
satisfy find
a
absolute constant
an
Boltzmann unit vector
statistics,
\jj^^
>
}\
of different
or
not. In
,
[Dys 4]
and if all the
and the total
Charge
particles
are
is zero, it is also
bosons
or
possible
to
such that
,
absolute constant [Dys 5]. Thus, in this
an
species
-Bn^f^
(^,H^)
T-
stable in any sense. One fermions belonging io q
case
the System is not
show, however, that if all the n particles are l different species, i.e., if one can partition the
can
>
arguments jcj, ,jc,j in the wave functions into q subsets such that the wave functions are totally antisymmetric under permutations within each subset, then the System is stable .
.
.
Hn>-Bq-'\ where B is
,
absolute constant
[Dys 4]. This is one of the most striking effects principle. It is even possible to prove stabihty of a Coulomb System of fermions of ^ different species, but all with negative Charge, interacting with particles of positive Charge, and arbitrary statistics. If the total number of particles is n it is again possible to establish an
of the Pauli exclusion
Hn where B is
>
-Bq-l^n
an absolute constant [Dys 4], (for the best constants and relativistic [[Lie 1]], [Lie 7], [Lie 8], [Lie 9]) for a simplified proof. It is not only the Pauli exclusion principle which is important for this result, but also the fact that charges interacting via the Coulomb potential tend to screen each other so that all interactions beyond the nearest-neighbor distance can be ignored. The starting point is a classical lemma proved by Onsager in 1939 [Ons 3]. If
cases see
Notes and Remarks
Rj
=
min
Xj
xj from its nearest
is the distance of
at
at Xi, then
states that
a Charge ej Onsager's lemma
|;c/
453
neighbor
among
n
charges
e/
n
> Z-^
All the results a
on
straightforward
stability
manner
\Xi
-
Xf
-r^
^^^j
of infinitely extended Coulomb Systems transfer in Systems with Dirichlet boundary con-
to bounded
ditions. The minimax
principle
in
in different versions in
proved by Weyl
in the
beginning of
Proposition 6.3.4 is a Standard result which is [[Kat 1]] and [[Rue 1]]. It was originally proved
the twentieth Century.
Section 6.3.2 measure was introduced by Wiener [Wie 1] in 1923. He was partially motivated by the work of Einstein and Smoluchowski on Brownian motion. This phenomenon is the earliest observed effect of microscopic particle structure and was first elucidated by Brown in the early nineteenth Century following a trip of exploration to Austraha. (It is probably the first physical theory with an Australasian infiuence.) The monograph by Nelson [[Nel 1]] contains a very readable account of the history and development of the various theories of the phenomenon. The construction of the Wiener integral that we have given is due to Nelson [Nel 2] (see also [[Ree 2]]). There is a large textbook literature on the Wiener integral from a probabilistic viewpoint. In particular the book of Ito and McKean [[Itol]] contains a great deal of information about regularity properties of the Wiener measure, including a proof of Theorem 6.3.6. The idea of the Feynman-Kac formula, Theorem 6.3.7, dates back to Feynman's unpublished Princeton thesis in 1942 (see [Fey 1]). He considered V^ is the Schrödinger equation rather than the diffusion equation. If HQ has the free Laplacian on L^([R^), one
The Wiener
=
|2
f^-itHö ^) W
ifij/
is
=
(4nitr'^
^^ exp{^^^}^(j) J>
sufficiently smooth, e.g., for i/^ 6 5^ (IR^). If F is a multiplication operator //o-bounded with relative bound less than one, one can use Trotter's
which is
theorem to prove that
(^-/r(//o+n^)(^^) =
lim
(
n-^oo\
)
n
)
/
7^v
/
7^v
d''xi'"d''xnQXp{iSn(xQ,xi,...,Xn,t)}il/(xn)
,
454
Models of Quantum Statistical Mechanics
where
Y^^
N
^,(XO,ZI,...,X,,O-E'^^
/
Feynman's idea
was
=
[l /|jc/-;c/_i|V
l
to rewrite the last
,/j
4 ^
\
limit
^
/
'^^
äs an
.
,
^
-^(^^0
/
integral
paths starting
over
at
^0,
(^-/^(//o+F)^)(^^^
^
/ Jco exp{z^,(co)}iA(a;(0)
,
where
^'()-/KS)'^^-/^(-(^^^ is the action of the
path CD. In this way Feynman viewed the Schrödinger dynamics äs a "sum over histories" of the particle's motion. Unfortunately, it can be proved rigorously that there does not exist a measure dco on the path space
Q/ such that this formula makes
Nevertheless,
one can
construct
sense,
for
see,
Feynman "integrals"
äs
example [Cam 1].
linear functionals
over
suitable function spaces, and this has been used to prove that classical mechanics is the ^ ^ 0 limit of quantum mechanics in a suitable sense, i.e., in this limit the Feynman "measure" is "supported" on the path of least action, which is just the classical path. The reader is referred to the monograph of Albeverio and H0egh-Krohn [[Alb 1]] for a review of these developments. The idea of continuing the Feynman formula to time that
imaginary
Wiener
measure can
be used to establish
rigorous [Kac 1]. a
Version of the
so
Feynman
formula appeared in Kac's 1951 paper Theorem 6.3.7 is a generalization of a version of the Feynman-Kac formula given by Reed and Simon ([[Ree 2]], Theorem X 68). The properties of the finite-volume Green's functions established in Theo rem 6.3.8 and the preceding remarks were established by Itö, Sato and Ueno, and Arima [Ito 1], [Sät 1], [Ari 1]. We have largely followed the 1973 paper of Angelescu and Nenciu [Ang 1] in the treatment of the finite-volume Wiener measures. In particular, these authors proved Theorem 6.3.9. The formulas for the Dirichlet boundary condition Wiener measures in Corollary 6.3.10 were used earlier by Ginibre [Gin 5] in his early work on the reduced density matrices in quantum statistical mechanics and by Novikov [Nov 1] in his discussion of the independence of the pressure from the boundary conditions. Example 6.3.11 is due to Novikov. A
comprehensive review of Wiener integral techniques in quantum statis prior to 1970 has been given by Ginibre [Gin 6] and the property of Wiener measures used in the proof of Corollary 6.3.10 can be tical mechanics
found in this article.
Notes and Remarks
455
Section 6.3.3
representation of the reduced density matrices in terms of Wiener integrals first given by Ginibre [Gin 5] in 1965. In particular, Ginibre derived the combinatorial formulas of Theorem 6.3.14, or really Example 6.3.15 since he dealt exclusively with Dirichlet boundary conditions. Similar combinatorics had previously been considered in the context of perturbation theory (see, for example, the article by Bloch [Blo 1]). This representation is discussed at length in Ginibre's review article [Gin 6] and a good short review of these matters can be found in [Gin 7]. The estimates for the finite-volume reduced density matrices with positive interactions, Theorem 6.3.17 and Corollary 6.3.18, were proved by Bratteli and Robinson in [Bra 17]. The low-density expansions used in proving Theorem 6.3.19 and Theorem 6.3.21 were developed by Ginibre using earlier low-density expansions derived by Ruelle (see [[Rue 1]]) for classical Systems. The bounds on the reduced density matrices given in Corollary 6.3.20 were not expHcitly stated by Ginibre but they are an easy consequence of his bounds for the
The was
associated correlation functionals. These bounds
are
of the form -l
'R\ß\zVf^ l l
\z
l''<-'l^ljJlTJl'-i TT
and for bosons
or
Boltzmann statistics
_
Pf^(of^)
>
0. These latter bounds have
generalized and systematized in the classical setting by Ruelle [Rue 16], following earlier ideas of Dobrushin. Ruelle proves that if the interactions are superstable, then the classical correlation functions satisfy bounds
been
Q
.
The method for establishing the infinite-volume limit of the pressure with repulsive (Dirichlet) boundary conditions discussed near the end of the section was developed by Fisher [Fis 1] and [Rue 12]. The method for deahng with elastic (Neumann) boundary conditions is due to Robinson [Rob 13]. It is elaborated in the monograph [[Rob 1]] and in particular the pressure is shown to be independent of the boundary conditions whenever the interaction potentials are positive and decreasing. Prior to Robinson's work the only boundary conditions that had been considered in the rigorous analysis of quantum statistical mechanics were Dirichlet boundary conditions. Robinson emphasized that the complementary use of Neumann boundary conditions, which were physically equally compelling, led to upper-lower bound techniques of great analytic Utility. These techniques are now commonly referred to äs Neumann-Dirichlet bracketing and have been applied in various other fields such äs constructive quantum field theory [Guer 2] and quantum-mechanical scattering theory [Marti 1]. For a review of these applications, see [Sim 2]. The functional integration method of proving independence of boundary conditions originated with Novikov [Nov 1] who considered hard-core bosons at low densities. This technique was reviewed by Ginibre [Gin 6] and further
456
Models of Quantum Statistical Mechanics
developed by Angelesen eontexts
and Neneiu
It has also been extended to other
[Ang 1].
(see [Sim 2]).
The infinite-volume limit of the pressure has also been established for some Systems for whieh classical stability falls, i.e., for different speeies of bosons and
fermions with Coulomb interaetion. We mentioned in the end of the Notes and Remarks to Section 6.3.1 that these Systems might have a type of quantummechanieal stability because of the Pauli excliision principle and the screening effect of the
potentials. The infinite-volume limit of the pressure for a System of positively charged bosons interacting via Coulomb forces with negatively charged fermions was proved by Lieb and Lebowitz in 1972 [Lie 6]. If the total System is electrically neutral, the limit is independent of the shapes of the finite volumes A, while the limit is shape dependent for Systems with a net charge. The existence of the
thermodynamic limit for two-dimensional neutral Cou lomb Systems was demonstrated by Fröhlich in 1976 [Frö 4]. Later Fröhhch and Park were able to use correlation inequality the ex techniques to prove
istence of the
thermodynamic limit for the pressure and the correlation functions at arbitrary temperatures and chemical potentials for Boltzmann particles interacting via two-body potentials of positive type [Frö 5] and they announced similar results for bosons. We will give a brief account of their results in the case of distinguishable particles with some simplifying assumptions. Consider a System of Boltzmann particles on U^' consisting of two different speeies of particles. These speeies are indexed by a "charge" q, where q may assume the values 1. The potential energy between a particle with charge q at the point X G [R^ and one with charge q' at x' G U^' is given by
U(q,X',q',x')=qq'^(x-x') where is
that it is
a real Symmetrie function on IR''. positive definite in the sense that
,
The crucial
assumption
on
<|) is
,z,, G [R^ For more technical
rea-
n
^QC,-)(x,--zy)>0 O' -l
for all finite sequences ci, ,c G sons one also has to assume that .
.
.
C,
(D(0) The
potential
for
n
particles
at the
U^"\q",x")^
xi,
.
.
.
< +00
points
Y^
.
x^^ and with
charges q^^
qiqj^(xi-xj)
is then
.
^
The Hubert space for region A C [R^' is
n
particles
A
=
charges
^i
,
.
.
.
,
^
confined to
(g)l'(A;^^T,-) =I-(A^V^"V') /=!
and the Hamiltonian is
with
given by
a
bounded
Notes and Remarks
^A ^(^'') where r(l)
7^^^
is
-
^SA(^'^)
457
Ä + ^^"H^"' )
=
V^ with Dirichlet boundary conditions. The partition function
is
now 00
_/J
.
,
Z^=Z^(^,z)=^^5:Trs(e-^<'(^)) '
=
0
q"
-E^E/^-"-''^''^^"H-^-") ,^iV^^" the
where
q"
finite-volume
runs
sum
e-^^A^(^'')(Jc^/)
over
all
(1)
'
and
possible charge configurations
2"
^-K'^^^").
The finite-volume pressure and the correlation functions are then defined by
is the kernel of
particle-density
Pf.(ß,z)=-^^\og(Z^(ß,z))
(2)
and
p^(ß,z;g",^)
{00
r
m
^^^/^^"^ jy"^-^^A"(^"^'")(xV'",xV")
m
=
Q^'
(3)
.
q"'
[Frö 5]. Adopt the preceding assumptions. follows for arbitrary ß > 0 and z > 0 and an arbitrary bounded regions A converging to R^ that
Theorem It
P(ß,z)
1.
=
of
lim PA (^, z) A
exists and is
sequence
^
IrS
independent of the particular sequence chosen, andP(ß,z) of the thermodynamic pressure'^
käs the
Standard proper ties
p(ß,z-q\^)
2.
=
\\m
pf,(ß,z-q",x")
A>oo
exists for
77
=
It is
0, l, 2,
monotonely increasing
in
(z(n/ßf'-~)\^p{nß<^((ü)/2} The theorem is
proved
via the
z
and bounded
by
.
Feynman-Kac formula,
which
now
takes
the form
^-K'^(.")(^^^,
^)
^
l
J(^\,ßT
1^ dtU(^\q\o,(tr]\J j^-A^(,,^)exp(l Jo
where
d|^^.:^l^(co")
=
l{d^,^;^;f(o^,]
.
,
(4)
458
Models of
Quantum Statistical Mechanics
One expresses Qxp{ J^ äs follows. Define a kernel W
dtU^^^'>(q^^,o}(tY}}
W(q, where
q,q'
^
at zero. We
{-\, l}]x,x'
identify
x'
q',
X, /;
G
by
of Gaussian
means
Integrals
by ,
U'']
t']
=
t, t'
W with the
qq'^(x [0,^]
G
-
x']d(t
-
t']
and d is the unit
corresponding quadratic
point
measure
form.
Define
=L-({-l,l} where
dq
is the
operator //
(1) (2) (3)
counting
on
X
r
[0,jö];^^x
X
{-1,1},
measure on
J^jcx
JO
.
and ehoose
some
selfadjoint
sueh that
H>^, //"" is Hubert-Schmidt for
H'^^WH''^ is
some 77 <
bounded form for
a
Let ^ consist of the real functions in
+ oo, < +
some
p|^>j /)(//"),
oc.
and
equip
^ with the
seminorms
/K.II//VII,
/7
1,2,....
=
The set ^ is then
a locally convex topological space, and condition (2) nuclear space. Condition (3) implies that W is continuous äs a form on 5^ x c^, and hence W extends to a distribution on 5^ (g) e$^ by the nuclear theorem [[Gel 1]]. But W is positive because O is positive definite. By Minlos' theorem [[Gel 1]] there exists a measure diJi^y on the dual
ensures
that 5^ is
a
space
^' of ^ such that
/
d^i^(^y^^(f^
=
^-(i/2)(/^/)
j^'
where
(fWf)
is the value of the
often called the Gaussian Wick ordered
with
product of the function .
(this
quadratic form
measure
mean
cp G
=5^'
is consistent with the definition of Wick
/ diiw(9]
n
:
e'Mfj}
6.3).
One
G
.5^.
ordering can
-exp
>
-
{f.Wfj}
l
By approximating
fj(q,X, t) with functions from 5^, and
9(f]
=
/
=
using
dq^q.d(x
-
(ßj(t]]
the notation
^,
t)f(q,
X,
in the
beginning
then compute that
V^
.
Ly-i
/l ,...,/
e^^^^^ by
i->
g/^(/). ^^(\m(fwf}^i(p(f]
the Notes and Remarks to Section
for
W on /. The measure /% is 0 and covariance W. Define the
t]dqd'xdt
of
Notes and Remarks
we
459
obtain
/\ diiw(9]
"
f^
{
n
'
/ -/o
^^P r
L
j\
^^^(^j^
^j
exp(-/^^t/('^H/,^ar)| l
=
^0
This is the desired Gaussian
l W' ^) r J
expansion
'
(5)
.
J
of
Qxp{- f^
dt
U'<^\q",co(tY}}
in terms of
Integrals.
Now, define
^A(^)-
E
/^^^/J^^.ß ^A^r/'^H^expjJ ^ ^^
E
/^^^/JüA,/, ^/^ri^'V^):cosf/^r(p(^,co(0,o): \J() J
^^{-1,1}-^^ -
{^1,1} VA
^e
.
and substitute the formulas 00
Zf,(ß, z)
=
(5)
and
(4)
into
r
_n
Combining
Q
this with
and
(2)
^A(;ß, z)
-
to derive
(3)
we
/ ö?/i^(,p)exp{zC^(
=
^'
=
(1) r
Y.- ^ d^i^(cp)c{((pr n
^^9(^,60(0,0^: J
.
^
obtain
^ log(y rf/.J,p) exp{zC^((p)}
and
pA(^,z;9",x")=ZA()S,z)-V X
One
jdn^(v) l[Jdß^:/^'^(coj) expl^i 1^^ dt
now
:
derives correlation
inequalities
for
:
exp{zC^(,p)}
.
quantities like
/ d^^((p}(p(fi}"'(p(fn) in much the
same
existence of the
way
äs
for classical lattice models, and
thermodynamic
limit
uses
these to show the
by monotonicity arguments.
One
can
also
extend the argument to establish the thermodynamic limit for other correlation functions and imaginary time Green's functions.
Section 6.3.4 The results
on
the
Böse gases with
Corollary
6.3.24
thermodynamic limits of the finite-volume Gibbs states for positive -body potentials obtained in Theorem 6.3.22 and appeared in [Bra 17], where these results were also proved for
460
Models of
Quantum Statistical Mechanics
general boundary conditions. The result alluded to in the Remark following [Par 1]. Results similar to Theorem 6.3.25 were proved by Duneau and Souillard [Dune 1] in 1973. Theorem 6.3.26 was proved in bits and pieces by several authors. The pointwise convergence of the Green's functions for C'" D was established by Gruber in 1968 [Gru 1]. (See also [Gin 8].) The füll result for fermions was proved by Ruelle [Rue 14], [Rue 15] in 1971-1972 by establishing uniform bounds on the derivatives of the finite-volume Green's functions and essentially applying Lemma 6.3.23, i.e., a technique similar to the one used to prove Proposition 6.3.29. The boson results were established with similar methods by Duneau and Souillard in the paper quoted above. Theorem 6.3.22 is due to Park in 1985
The
study of the infinite-volume limit of Green's functions from a general viewpoint originated in a 1970 paper by Dubin and Sewell [Dub 1], soon to be followed by Ruskai [Rusk 1] and Sirugue and Winnink (see [Win 3]). Some later refinements were made by Bratteli and Robinson [Bra 16] and J0rgensen [J0r 2].
In
[Dub 1], [Rusk 1], [J0r 2] the multitime Green's functions are viewed a certain *-algebra essentially introduced by Wightman in 1956 [Wig 3]. This is the free *-algebra ^(^) generated by pairs of elements (A, t)
äs
States
on
with A e^ and
t G
U modulo the relations
(^1,0(^2, 0 l(^l,0 + (^2, 0
=
(^1^2, 0
=
(^-4i+^2, 0
.
.
and
(1,?) and ^1,^2 e
2l, A
linearly spanned,
e
C,
(1,0)
R. More specifically, let SB denote the *-algebra complex numbers, by strings of elements
? e
the
over
=
P=(Ai,ti)(A2,t2)---(Ak,tk) with AI e
9l,
t, e R. Define the
P* and let
=
*-operation by
(AI, tk}(Al_ tt-i)
-
(A\, ti)
3 be the two-sided ideal generated by the elements
(AI, t)(A2, t)
l(A^,t)
+
-
(AiA2, t)
,
(A2,t)-(lAi+A2,t)
,
and
(1,0 -(1,0) Then
S(^)
group 7 of
is the
quotient *-algebra 2B/3. *-automorphisms of 35 (^) by
7,((^i, ^i) One
can
then prove the
[Rusk 1], [J0r 2].
(A-, 40)
=
.
One next defines
(Ai,ti+t)'" (Ak,
following analogue
tk +
a
one-parameter
0
of the GNS construction
[Dub 1],
Notes and Remarks
Let
Proposition. P G 33 (^))
(i) (ii)
, U
on
co(yt(P]]
be
co
33 (^) =
and
^(P]> for allteU and P e 23(^). t\-^o}(P*yt(P)) is continuous on U for exists
*
a
anda
all
for
-representation
strongly
n
continuous
ö/S(M)
all P e
on a
(^).
Hubert space
unitary representation
U
of
co(P)-(Q,7r(P)Q), ^(00 Q, n(y,(P)\ U(t)n(P]U(-t] -
=
ifwe identify
^
äs a subalgebra of^(^)byAe^^(A,0)e S(^) ofnto^ itfollows that
and
cD((A,,ti)(A2,t2)'"(Ak,tk)) =
(Q, nQ(Ai)U(t2
-
^1)710(^2)
U(tk
Furthermore, conditions (l)-(3) determine
proposition gives no Information through the formula
This S (9l)
0
>
the conditions
The mapping
UQ is the restriction
(4)
positive linear functional (co(P*P)
satisfying
Itfollo\vs that there a cyclic vector Q G on 9) such that (1) (2) (3)
a
461
on
-
n
4-i)7ro(^^)Ü). and U up
to
unitary equivalence,
which functions G define states
on
G(Ai,. ..,Ak]ti,..,,tk)= o}((Ai,ti)(A2, t2)"' (Ak, tk)) and Theorem 6.3.27
gives
the first
complete
characterization of the
two-point
functions for this to be true, although [Win 3] and [Bra 16] contain partial characterizations. The technique of dilating the map t\-^Xf io a unitary group
t^-^Uf used in the proof of this theorem goes back An
exposition less complete,
of this and other dilation theorems in
be found in
[Nai 3]. [[Eva 1]] and,
separating character of the vector in Theorem [Win 3] and the example in the subseqUent warning is due to Kishimoto. The reader should be aware that Theorem 2 in [Bra 16] is erroneous, due to disregarding this warning. The semi-abstract setting discussed prior to Proposition 6.3.29 was introduced in [Bra 16], and this pro position and the associated corollary and remark are taken from this paper, although very similar results occur in [Win 3]. The main theorem in Section 6.3.4.2 on Böse gases with repulsive interactions appeared in [Bra 17], and all the estimates in the subsequent lemmas are taken from [Bra 16]. 6.3.28
was
[[Rie 1]].
The
to Naimark in 1943
can
demonstrated in
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a
two-dimensional
Ising
List of
Symbols
Standard Notation
^, 93, G: 50^5 ^5 3
von
3
Ideals in C* and
A,B,C
elements in C* and
co, 9
States
T, y, a,
ö"
C*-algebras Neumann
algebras
automorphisms
,5^
Hubert spaces
von
or
Neumann
von
groups of
a
vectors in Hubert space
G
a
group
A
a
bounded
v
automorphisms
*-morphism
71
Q, (^, ?7, i/^ P, E, F
/x,
algebras algebras
Neumann
orthogonal projections region
in
a
Euclidean space
or on a
lattice
measures
Special Symbols Roman numerals I and II refer to Volume I and
numerals refer to the page number of the first of the Symbol.
\A \ 1 1^1 1
II, respectively, and arabic
occurrence
in the second editions
Absolute value of ^
134
C*-normof^
120
488
List of
Symbols
A+, A-
Positive and
^*
Adjoint
A-^
Inverseof^
I 19
Gelfand transform of A
A
Affine functional defined
[A, B] {A, B] (A, B) A(K)
Commutator
=
AB
Anticommutator
-
^
I 63
by
^
I 339
BA
AB + BA
Canonical correlation
II 94
The real affine continuous functions on
A(G] A(p], a(f) a*(f)
I 35
125
A
^
of A
negative part
of^
the compact
The almost A* (p)
set K
convex
periodic functions
I 321 G
II 36
Fields in momentum space Annihilation operator on Fock space Creation operator on Fock space
11 448
over
<7^(/)
Annihilation
a+(/), -(/)
Bose/Fermion annihilation operator
or
creation operator
11 8 11 8
on
Fock space
on
11 9
Fock space
II 9
+(/), !(/)
Bose/Fermion
flf (/)
Bose/Fermion annihilation
a(x)
creation operator on Fock space Annihilation field operator valued
a*^(x)
Creation field operator valued
floj(/)
Annihilation operator in representation
a*j(f)
Creation operator in defined by CD
ax,a*
Spin lowering
creation operator
on
Fock space
II 9
or
II 9
distribution
II 10
distribution
defined
by
II 10
11 25
co
and
representation II 25
raising operators
II
258, 425
ä
Unit extension of ^
I 23
^1+
The
I 32
positive part
of ^
n
(X)% /
Finite tensor
product of C*-algebras
I 144
l
=
(X)^la
Infinite tensor
^(X)^G
C*-cross
(^l, G, T) (^l, T) ^l_ (l)) ^(W ^1+ (H) ^r(//)
product of C*-algebras
product
1138
^IA
C*-dynamical System C*-dynamical System with G=U CAR algebra over E) CCR algebra over H Entire analytic elements for T Fixed point algebra under action Local algebra over A
a.,
Dual action of a/
oc^(co)
Characteristic function
5v)i
The
,
,
"il^ ^r
norm
less than
I 136 11 76 11 1 5 II 20 II 76 T
II 240 I 140
on
path
positive linear functionals
with
I 145
or
equal
space
on
to
II 377
^l one
I 53
List of
B(f] B(i)
Fermion field operator Fourier coefficients of B
B^ BX
Banach space of interactions Banach space of translationally
b(ii] 93 (^)
Barycenter of ^ Free algebra over
ß ß^ ß^
Inverse temperature
CQ(X]
The continuous
Symbols
489
11 17 II 206 II 242
invariant interactions
11 297 l 323
^
II 460 II 5
Critical inverse temperature Inverse scattering morphisms
II 331 II 164
C(X]
complex functions locally compact Hausdorff space X vanishing at infinity The continuous complex functions
Cij(X]
The bounded continuous
C*(^,T)
C*-crossed
Co
Weakly continuous Weakly* continuous
on
the
I 2l on
the compact Hausdorff space X functions
CQ CO(IG(A)) C"^
on
l 24
X
1138
product
The
convex
The
m
hüll of
times
I 164 I 164
{T^(^); ^
continuously
able functions
or
G
G}
The
Cfin(^ß)
Continuous functions
I 381
differentiII 37
states
C
complex
121
complex
numbers
depending
on a
finite number of coordinates
II 367
C^(G) C^(G)
Functions
generated by expectations
11 202
Functions
Q
generated by expectations asymptotically equal to co Lattice configuration
II 206
States
II 282
(0 (A
Centralizer of
co
II 1 1 3
Local
of classical lattice
(A
Element of isotonic
D(S) D(G) (Dij/: Dq)\ D(X] dY(H] d(y, A)
Domain of linear operator S Domain of graph G
c
algebra
II 240
System
Connes'
family
Radon-Nikodym cocycle
Diameter of X Second
II 329
quantization
II 409
I 186 I 147
11 242 map
11 8
Distance from jy to A
11 374
d^l^^,^^ym^,(üf^,a)'^]
Measure
II 384
d(ß
D^
Path space Strip in C
D^"^
Tube in C"
A
Modular operator
A^'i^<^2
Relative modular operator
A^^(^)
Root
on
mean
composite paths
measure
II 386 II 8l II 150
square deviation
I 89
1109 I 349
490
List of
A(D
Symbols Set of
invariant
translationally
maximal entropy states v
-v-
TheLaplacian
-V^
=
j
(
ö(j} ^A ^(D
Unit
df(K)
Boundary
dF:,
Derivative of F at
ÖA
Boundary
E^i
E^
State space of ^ The G-invariant states
E(a
Projection
II 49
nv
^^i
/-l
point
II 294
^
I 322
measure on co
Generator of local evolution
11 244
Derivation defined
11 259
set of
O
by
/
I 326 U 297
x
of A
11 55
I 53
51
on
I 374
t/f^( G) -invariant
onto
vectors
El^ Ey
E^^(x) 'A E(p
I 381
The invariant state extensions
II 201
Projection onto eigensubspace of regulär representation Energy of interaction at point x Projection onto local subalgebra Mean energy at 0 for a translationally
II 242
11 208
II 246
invariant interaction O
:
e'^^(^):
Wick ordered
11 291
exponential
11 458
elf
Matrix units in CAR
e((x,/x)
Matrix units in lattice Systems
<^(K)
The extremal
points
algebra
of the
II 16 11 306
convex
set^
(f
1318 II 390
(ß,z)
Banach space of path space functions Local energy per unit volume
e^
Combinatorial factor
11 385
F
F
Fgi
The set of factor states
F^
F,
FA,B(t) F(CO),FA(CO)
=
II 53
J^-^'^
=
189 on
^
I 372
[7i(2I')',]
FA,B(t)
=
I 383
o,(Ai,(B))
1181,178
Free energy functional
on
matrix
algebra,
11
90, 269
F^((D)=ß-^S^((D]-(ß(H^(\]) F^(ß^)
Free energy
on
matrix
algebra
II 269
FA(^a))--r'logTrA(^-^^'^(^)) Fß^^(o})
Mean free energy
F/^o(co) F(^)
=
S(o})
Thermodynamic
functional, -
11 292
ßH^(o})
mean
free energy
11 292
List of
/(co)
Upper envelope
{/, g} /(D(^) 5({))
Poisson bracket
Fock space
over
'S(V)
Bose/Fermi
Fock space
(g, ^, G, T, 7, a)
Field System
g_(.(A)
Bose/Fermi
Böse field operator on Fock space Böse field operator in representation
,
.
.
.
,
I 326
/
II 232
functional to
Tangential
mean
jc^O
11 302
free energy
11 7
I) over
11 7
I)
II 197
Fock space
over
L^(A)
given by co Spin System interaction Norm on spin interaction Norm on spin interaction Translationally invariant two-body Potential ^-body potential
^(X) 1 1 Öl l ^ ||>||; O(jc/ Xj) O^^') (jci
of
491
Symbols
O(;c)
Free Euclidean field operator valued
^(x)": (/)(^' ^)
Wick ordered power of field
G(S)
Graph
G
Dual group of the abelian compact group G
II 355
II 12
II 24 II 241 II 242 II
247, 252 II 358 II 358
distribution :
(^(^' ^)(^)
-
11 448 II
II 206
(p(Ayg(B))
of the linear operator S
GA, B(t)
GA, B(t]
G(p
Stabilizer group of cp
=
o}(it(A}B)
-
||öf||;^ Y(m] F(a) F([/)
f
I 186
locally l 249
co(A)(D(B)
II
83, 178 11 202
GA(^o,...,An^i,...,^.)l GA(^"^ O Gf,(A,B',t) G(A,B]t)
448, 458
11403
Finite volume Green functions
J
II 403 II 404
Infinite volume Green function
II 404
Covariance of free Euclidean
II 448
measure
Invariant F of 9Jl
I 149
1310
(y, t)
F-spectrum of a Second quantization map Unitary cocycle Unitary cocycle defined by perturbation P 7 G G acting on ^ G G
JB
yB(A]
7
Scattering morphisms
II 160
Contour in lattice System v-1-dimensional area of contour
II 321
F^
Ff
7
l
y
//A
l
Seif
H^ //
B^AB
adjoint
on
H^
=
-
II 8 II 127
the II 145 I 250
{B^B,A}/2
extension of the
11 9l
Laplacian
L2(A)
Laplacian Laplacian Laplacian
11 322
n 49
with Neumann
II 55
with
II 55
with
conditions
boundary conditions a-boundary conditions Dirichlet boundary
II 55
492
List of
//P^^
Symbols
Laplacian
with
periodic boundary
conditions
II 55
HA,B(t} //O (A) ff (A.)
HA.B=FA,B-GA,B
11185
Finite volume Hamiltonian
II 241
H^(co)
Mean energy functional
translationally invariant state co (H) Macroscopic external field strength
U 291
(H)
Condition
II 410
//()) //c,^A
Ground state energy functional Total finite volume Hamiltonian
II 347
Hl"\
w-particle part
of
II 357
fi
Planck's constant
h(x)
Microscopic
/z^'^^
Form associated to
(,71) (,7i, Q)
Representation
H
Locally
correct Hamiltonian
II 245
the
on
II 318
II 356
//^,A
I 4
external field
strength
atjc
11319
H^'^
11 361
143
on
Cyclic representation
on
with
g
cyclic
vector Q
I 45
(^, TCcü, QO.)
Cyclic representation
(y
Hubert space in defined by co
(X)^
Finite tensor
state
defined
by
the I 57
(D
cyclic representation I 57
product
of Hubert
'^^
I 143
spaces
(X)^"Scc
Infinite tensor
A
Bounded domain Hubert space Conjugate Hubert space of g
l)"
w-fold tensor
^'!^,^"_
/
1
product
of Hubert spaces
I 144 II 240
I 70
product of l) with itself Symmetrized/antisymmetrized part of l)" Identity Identity
on a
of
an
11 7
II 7
Banach space
operator algebra
I 2l
3^
I 250
^
Identity automorphism algebra
J
Modular
y^i,^=2
Relative modular
J(A)~
j(A)=JAJ
j(x,y)
Ising interaction
K^i Kß
The set of KMS states at value
A^oo
The set of
5^(^, G)
Continuous functions G
of
operator
an
conjugation conjugation
I 89
1109 1102 II
Modified Hamiltonian,
ground
K^,
=
states
with compact support
H
-
ß
]^
257, 319 II 46
II 1 13 II 1 3 1
h^
^ I 137
List of
LP(X, G]
493
I 341
K,(f} LP (PL], LP (X,^]
Symbols
L^-spaces of complex functions The L^-functions from the locally
183
compact group G into the Banach
space^
1138
L^(^o) L^(co)
Notion of
II 160
Work in
11 213
L
Lattice
L\^c(^'\S]
Local
^(X]
The bounded linear operators
J^'^() J^,() lAl A^
asymptotic abelianness cyclic process
II 239
L^-space
II 360
Banach space X The compact operators
The
predual
Area of A,
of
or
MJ^(K]
The
The
M(A) M(ß, hx)
Meanof^
The
probability probability
center
M
M(/z) 9Jl',9[R''
Radon
compact
M\(K] MCO(K)
I 2l
I 69
number of
;.
positive
I 20 on
^(S)
Complement of A Thermal wavelength,
convex
the
on
l
-
points
in A
(47ij5)^/^
measures on
II 53
the I 321
set K
measures on measures
I 321
Ä^
with
baryl 323
CD
1383
Mean
magnetization of atom at point Overall magnetization Maximum mean magnetization respectively,
of set 9[)1
The closed linear span of 9[)IR, projection into this
9Jl,
The
II 318
II 335
171 or
y)r
predual of 9Jl j^*-dynamical System pr*-dynamical System with G=U l^*-crossed product The fixed point algebra under the
^J^T:
The entire
aR (g)^ G
11 314
Commutant, and bicommutant
[9315^]
(50l, G, T) (9JI, T)
x
action
action
the I 72 I 75
1136 U 1l I 139
11 113
T
analytic
elements for the I 78
T
A''
(X)9[R/
Finite ['f *-tensor
product
I 144
/=!
(X)""9Jia
Infinite r*-tensor
%
^E
/l V
v
/z A
v
/i :^ ^ >^ I.L
=
product
E^RE
Least upper bound Greatest lower bound
v
v
Choquet ordering Bishop-de Leeuw ordering Chemical potential
I 144
l 389 I 323
I 323 I 325 I 458
11 5
494
List of
Symbols
Spectral
^^
measure
of
automorphism
group in invariant state ^^
KMS-measure
l^c
Central
J.LE
Measure of
J.LX
Maximal
/,if
Conditional Wiener
^
II 88
Esir
on
11117
11117
measure
ergodic decomposition
11120
E^n
11 131
measure on
11 368
measure
I.L^
Wiener
/.fj'"^' ^
Conditional finite volume Wiener
1.1
j.i=(-V~+m~Y^^
measure
/,i^
II 369
measure
with
boundary
conditions
11 375
er
11449
Finite volume Euclidean Gibbs
^
11 449
measure
I.L^^,
Gaussian
N^ii
The normal states
^
with square
measure
on
the
mean
II 458
C*-algebra
A^^^
Hubert space The G-invariant, co-normal states
A^
Number operator
Nf ^c.,/
a^(f]a(f] Nf A'a.,/-<(/K(/)
NO)
Number operator
N((D] N\(o})
Number
A^A
Local number operator Set of nearest neighbours in A
N
W
I 131
on a
on
on
^
I 385
Fock space
11 7 II 26
=
N(A)
1127
on
^
functional, 7V(co)
11 3l =
co(Mo)
H 37
Local number functional
n,,^f(i^)
/loj
Form associated to N^o
/U
A
VA
Spectral
=
II 321
\\a,,(fm'
no^j-
=
II 52 II 363
1127 II 30
^a;a^ .r
6
II 425
A measure
of
automorphism
group in invariant state
0(0 o(t) (^f^('sii), (9^^
Order
II 88
l 202
t
Smaller order than
l 202
t
The
orthogonal probability measures Eyi with barycenter o) Cuntz' algebra Cyclic vector associated to the state co
l 340
on
(9 n
Qf,j Q^
Path space Space of Holder continuous
Q^ Q ß
II 149
II 367
paths
II 368
Space of bounded paths
11 375
Vector state associated to the vector Q
coi J_ C02
Orthogonal states Disjoint states
i
0)2
II 369
Path space
coß
o; l
I 57
Perturbed vector
Q/j
QA,
II 110
co/7
Fock
(jüT
Truncated state
vacuum
I 48 I 340 I 370
state
II 44
11
39, 43
List of
coA,
CD^
Local Gibbs state
II
O)
Normal extension of
0)
Extension of
co^
Perturbed
(KMS-)
(DA
Quasi-free
state
co^''^
Local Gibbs state for
co
to
to
oj
na)(^)"
Ä
cof
^
of
Family
P(a
Projection
m
50, 6l, 263, 377 II 80
11 149
state
II 219
interacting II 381
Infinite volume limit of
co'"
495
II 116
quantum gas '
Symbols
cof ; {^
II 38 1
trajectories
II 384
onto
(7öj-almost periodic
onto
even/odd subspaces
1 408
vectors
P
Projection
P(ß, z) /^
n 320
P($)2
Thermodynamic pressure (l (r])/2 P^ Polynomial local interaction
p^(x)
p^(x)
Y,\\^(X)\\
II 244
of Fock space
II 7 II 53
=
=
11 449
X3X
(e^^"](x,y]
p(x,y',t)
p(x,y,t]
Pc.^(^.y\t] /'per 'A (^' y^ 0
Pa^^(x.y\t]
^,^Q
Natural
^7
Partition of / into ordered subsets
II 39
n(/)
Böse field operator
II 12
=
=
II 367
(e-
Green function with
boundary conditions positive cone on
II 380 I 102
Fock space,
n(/)
=
cD(//)
The
*-morphism in the cyclic representation defined by co Unitarily equivalent representations Quasi-equivalent representations Disjoint representations
Tiaj
TCi
~
7l]
^
71]
II 373
periodic
A
712 712 7i2
R(S) R(G)
Range Range
[R
The real numbers
[R+
The
of linear operator S of the graph G
PA
non-negative real numbers Spectral radius of A Particle density Finite volume particle density Critical density of Böse gas Density of Bose-Einstein condensate Finite volume reduced density matrix Density matrix of finite volume Gibbs
PA(W'") PA (ß, ^; ^"
Density functional on trajectories Finite volume particle-density
p(A) p(ß,z) pf^(ß,z) Pc(ß] Po(j^)
P^ A(/"5-^'")' PA(y";^"0
state
,
^^ )
correlation functions
I 53 I 48 I 80 I 370
I 186
l 26
II 52 II 56
II 64 11 69
11 382
II 383 II 384
II 457
496
List of
Symbols
JA^/^
S
5
^c, c. 5(911) S(K}
^c,
set K
I 321
Sx(U) S^^(i:(A)) -S'A(CO) 5(^1 v) S/^(o}\(p)
Convex combinations of U (g)' s
l 382
^A,A'H ^A(CO)
^A^A^co) ^^^^^.(co) -5;(co)
II 274 II 276
S(coi cü?)
Conditional entropy Relative entropy of faithful normal
S(co)
Mean entropy functional
*S'^(a;) s(f^g)
Classical action of the
,
=
,
c.
=^
189
^c,
,
c.
^Jf
Invariant 5 of
^
I 1 09
c.'
I 149
The real continuous on
the compact
convex
convex
Convex combinations of
Entropy functional
on
functions
i:g(Ays
matrix
l 382
algebra
II
89, 267
Relative entropy of positive measures Relative entropy of states on a matrix
11 267
algebra
,
II 268
States
II 278 trans-
invariant states
lationally Real
on
II 288
path positive Symmetrie bilinear form
II 454 II 18
4"A(iA)
Boundary
o^i(A) a(^) (jj^
Spectrum of J e ^ Spectrum of the abelian Modular automorphism
o(X, F)
Topology
on
defined
by
(Ju(Y) (j(U)
?7-spectrum of Spectrum oft/
ffp(U} (TP(T:)
Point spectrum of U Point spectrum of T
ö-(/, g)
Real, non-degenerate, symplectic
Ö-;
Pauh matrix
J d^(z)^(z) / ö?^i(z)(z) J di^(z)T(z) /J* d^L(z)'^l(z)
Direct
integral
of vectors
Direct
integral
of Hubert spaces
I 441
Direct
integral
of operators
I 443
Direct
integral
of
von
f diJ.(z)n(z)
Direct
algebras integral
of
representations
term in
t^^\
II 356
'
associated to
I
C* -algebra ^
161
group I 96
co
the linear space X the subspace F of the dual
I 97
7
I 251
1251 l 407 I 407
bilinear form
0a
0^
Direct
sum
0
TTa
Direct
sum
II 19 I
T)^'
Self-adjoint
ff,
1441
Neumann
of Hubert spaces of representations
Critical temperature
p' Al
369, II 241
I 445
TC
in
25, 28
extension of the
1447 I 46 I 46
11 318
Laplacian
CO^(A)
n 355
Self-adjoint extension of the Laplacian corresponding to cr-boundary condition
11 356
List of
T,^^
^(T,A rr(n}
=
dr(Tl]\)
^(T,A T-i-U
t((p, lA) tW
Sesquilinear form Quadratic form
of
-particle part sum
T^^ A
II 357
of semi-bounded operators
11361 II 26 II 27
r' >0
d:^A(^)
Form associated to
^() T(jr,F)
The trace class operators
II 356
C7,A
(T, ^)-KMS
Mackey topology Finite volume dynamics Equilibrium condition
f
Normal extension of
T
168
on
I 163 II
cx))-KMS
49, 247 II 77
to weak
closure
(T,
497
II 356
The
Form
Symbols
II 80
Ground/Ceiling Extension of
T
state condition
to
1197
^
II 116
<
Perturbed
^free f ^int
Free time evolution
II 23 1
Time evolution with interaction
II 23 1
''/ ^^
automorphism
II
145, 212, 218
Time evolution defined
by Potential t/co
group
II 254
Unitary representation the G-invariant state
157
co
fdt,(t)U,(A)
f/.(^)
U,(A)
t/A
Finite volume
=
associated to
potential
I 165 energy II 356
operator
^r
The
U("\Xi,...,Xn)
Function
U((D^] U(d]
Interaction energy of co'"
II 384
Cutoff
II 449
U^"\q",x")
Potential energy of
charged System
II 456
4"^(^)
Form associated to
t/|"^
II 360
%(^) ^(^) ^(y4,)
Connected component of 1l of The unitary group of 5l
of L'A
-particle part
defining
II 357
L^|"^
II 357
P(<E))2-interaction
Neighborhood
^(^)
of ^
II 101 II 101
120
Propagation velocity of physical disturbances
^*(9[,T) ^(/) r
II 260
^*-crossed product Weyl operator M011er scattering matrices
II 164
II 249
I 139 n 13
^o(A)
Surface energy
^(i>(A,A')
Interaction energy between A and
2B
Free
algebra
II 460
Z.
Predual of ^
197
A^A
II 263
List of
498
X^(E), XQ^(E)
Symbols Spectral subspaces corresponding
to
E CG
1251
^f
Chemical
SA
Classical
potential subgroup
of gauge II 199
group
X
partition function Magnetic susceptibility
ZA,ZA(JS, z)
Partition function,
ZA(JÖ, z)
II 365 II 318 =
11 94
rr^(e-^(^A-^'^A)) e^^'
z
Activity,
Za^\(x,y,t)
Compensating The integers
Z
Z^'
z
=
II 47
Greens function
The v-dimensional cubic lattice group Center of^Jl
3(9Jl) 3,0
II 374
II 239 171
Center of 7l,, (^f
1122
3^
Commutant
I 122
3^
Algebra
Ill^lllg
Norm
i/^(/i,
.
.
.
,
fn)
at
on
w-particle
algebra infinity
I 122
interactions vectors
on
Fock space
II 418 11 9
Subject
Index
Page numbers
in
to this
plain type
refer to the second edition of Volume I, those in italics refer
edition of Volume II.
Abragam
and Proctor
Absolute value
220
Banach
34
19, 20, 25, 137, 152, 430
Borchers
Action
C*-
continuous
463
3, 19, 20, 25
393, 400 dual 140, 149, 375 ergodic 137, 142 free 137, 142
commutant
finite
von
transitive
fixed
point
464
Activity
function
442
hyperfinite
19
440, 441
Akemann
460
Akemann and Pedersen Albert
309
H0egh-Krohn
449
8
Algebra abelian
19, 35, 58, 62, 83
antiliminal
352
approximately 152
finite
233
et seq.
147
310, 313
2l 249
von
von
Neumann
Neumann
infinity
150, 151
147
122, 123, 126, 156, 318, 370, 373, 440, 453, 454, 205 Jordan 7, 16 ^() 20, 38, 68-72, 85, 96, 106, 124, 129, 131, 225, 226, 245-248, 258, 292, 305, 306, 309, 321, 356, 379, 398, J^^() 2l, 72, 129-133, 246, 247, 306, 307, 309, 352, 4, 5 local 1l, 118 at
7
Albeverio and Alfsen
group
infinite
Aizenman
5*-
Neumann
47, 63, 64, 70, 383, 384, 389-392, 395, 401-403, 421, 429,
Adjoint
et seq.
122, 126, 370 commutative 19, 2l, 6l, 154 direct integral 445, 446
multiplier
309
500
Index
Subject
Algebra (Contd.}
and Shiraishi
maximal abelian
Neumann
320, 350, 361, 367, 391, 417, 420, 447, 448, 452 von
-dimensionally homogeneous
351,
459, 460 normed
220
and St0rmer and Woods and
Wyss
Arima
449
231 220
220
Arnold and Avez
20
Arveson
nuclear
144, 151, 232 operator 3, 136, 439
prime 310 properly infinite
von
Asymptotic abelianness 14, 340-343, 380, 406, 414, 461-463, 40, 43, 144,
Neumann
159, 163, 174-183, 195, 198, 210, 240, 303, 320, 326, 422, 423
135,
141, 147
purely infinite von Neumann 147 quasi-local 1l, 8l, 118-129, 133-135,
Asymptotic abelian L^ (U)- 163-168, 176-181, 188,
Australia
reduced
Auto-correlation
Neumann
self-adjoint semifinite (T-finite
155
19
von
Neumann
von
in
mean
403, 53, 124, 126,
442
bound(s)
9l, 92, 94, 96, 108,
109, 113, 159, 227, 283, 286, 333, 437
bound(s) 94-97 Automorphism 12, 13, 16, 44, 56, 96, 97, upper
142
102, 108, 139, 209-313, 37^M52, 4, 15-19, 40-43, 5l, 57-63, 73-85, 101, 109-113, 189, 195, 214, 243297, 306, 321, 331, 414, 436
151
UHF
(Uniformly hyperfine) 129, 156, 236, 268, 306, 307, 308, 351, 353, 405, 17, 229, 239, 246 von Neumann 3, 65, 70, 75, 76ff. W^3, 155, 305 Weyl 30, 35, 37, 38 Algebras, measurable family of Analysis, abelian harmonic 140 Analytic continuation 25 Analyticity strong 99, 155, 178 weak
403, 132 ...
234
147, 148, 150 84, 151, 155
semisimple 306 simple 7, 24, 44, 133, 134, topological 7 type IC*-
weakly weakly
lower
Neumann
198-
201, 236
156, 219, 289, 317, 321, 352, 373, 450, 453, 454, 460, 5, 16, 22-25, 4l, 63, 118, 177, 239, 240, 244, 259 quotient 39, 40, 4l, 43 von
463
152, 308, 309
445
anti-
209,211,221 141, 148, 149, 225, 249, 283, 293-295, 300, 305, 310, 312, 313 Jordan 211,212,218,219,222,224227, 229-231, 277 inner
modular
(see Modular automorphism)
Order Avez
211,225
464
90, 155, 178
Angelescu
and Nenciu
450, 452 408, 412, 417, 429, 436, 455,
Annihilator
22, 204
431, 432
and Kishimoto
and Lieb
and Sewell
442
Baire
(see Measure, Baire) 318, 322, 324, 331-333, 350, 353357, 367, 458, 137, 138, 367
measure
Approximation theory 202, 290, 299, 304 Araki 152, 155, 156, 279-222, 22^-229, 427, 432, 436-439 Araki, Haag, Kastler and Takesaki 234 and Ion
Babbitt and Thomas
234
435
222, 436, 437
set
structure
Balslev
279
Banach
154
352
Barycenter 318-350, 359-390, 424, 426, 431, 433, 440, 450, 458, 777, 779, 135, 306
Subject Base
(of
a
334
cone)
Bauer maximum
principle
Baumann and Jost
331, 293, 301
435 227
442
Bicommutant Bidual
72
130, 217
Billingsley
435
233
Bishop Bishop
and de Leeuw
Bloch
453
458, 459
218, 221
automorphism
797
435, 436
4
302, 309 Borchers 308, 309, 311, 312 Borchers and Yngvason 463
Borel
304,311,425
Kishimoto and Robinson
255, 442 305-307, 311, 455, 459 Brownian motion 366, 376, 448 particle 367,368,377 424
Bucholz and Roberts Butzer
288
304, 311
304
Cannon
section
304, 3 1 1
233
and Kishimoto
Brush
Boost
automorphism
229
31 1
and Robinson
3,430
map
Haagerup
and Kastler
compactification
function
229
Herman and Robinson
93
7(5-27, 47-57, 73, 77, 99, 148, 161, 164, 218, 219
cross
54
Bratteli
and
437
product
Boltzmann
151
law
Elliott and Herman
transformation
Bohr
53, 55-57, 373, 376, 389, 393, 451 periodic 55, 64, 380, 444 Boundary set 326, 327, 330, 331, 353
Elliott and Kishimoto
and Khatset
Bohr
Neumann
Boyle's
436
Phelps
Bogoliubov scalar
56
444
Bourbaki
Birman and Kato
and
elastic free
Berman, Bruch and Fortune Bethe
501
379, 386-393, 396, 401, 402, 414, 427, 444, 448-455
231
Batty
Index
270
Cantor-Bernstein
295
argument
5, 295, 187, 377
(see Map, Borel) (see Measure, Borel) 5,289,295, 318, 322, 331, 333, 338, 340, 343, 350, 354-357, 412, 458, 460, 463, 186, 367, 377, 379,
set
8l
306, 308
CAR
measure
relations
set
algebra 14-16, 23, 37, 42-53, 77, 99, 108, 148, 161-169, 219, 225, 237, 350-355, 395
380 structure
Carnot's formula
350, 440, 446, 362
Böse gas
14, 5^-75, 797, 795, 275, 220, 257, 353-420, 442, 459-461 Bound, greatest lower 323, 334, 335, 369, 385, 777 least upper 76, 78, 79, 121, 323, 369, 385
27^
Cauchy estimates
178
representation transform
Cayley
242
25
Numbers
7
CCR relations
Boundary conditions
160, 3, 48, 63, 74,
146, 147 57, 54, 62, 355, 363, 392, 444,
classical
446
completely absorbtive completely reflecting Dirichlet
6, 10, 77, 74
376 376
55-57, 64-70, 356, 363, 374-
6, 10-15 75, 75-54, 40, 57-60, 67, 218, 225, 237, 349, 354, 355, 395, 396, 402, 443 Center 7l, 123, 125, 126, 147, 295, 317, 364, 370, 389, 390, 445, 446, 453
algebra
ofMass
318
Central sequence
309
502
Index
Subject
Centralizer Chaiken
61-63, 139, 249-251, 351, 407, 430, 464, 130, 131
Chemical
potential 5, 5, 45, 197, 198, 204, 210, 211, 234, 396, 410,
Chernoff
303, 304
Circled subset
of factors
224
98, 163
Classification of
C*-algebras
145, 151
145
118, 128, 156, 393, 398, 400, 401, 404, 406, 420, 429, 438, 440, 42, 106, 118, 163, 178, 185, 194, 198, 201, 218, 423, 437 402
C*-norm
20, 22, 3l, 143 Cocycle 141, 298, 312, 131, 145-148, 155, 157, 212, 287, 316 identity 294, 297 294, 296, 299 296, 297, 299 312
Cohomology
of [R is trivial
433, 445, 1l Commutation relations 4
6
Commute
strongly
278, 279, 107, 108,
116, 191
Condensation, Bose-Einstein
63, 64, 70, 74, 78, 353, 354, 403, 440
Conditional
97, 98, 100, 164-184, 194, 202, 204, 219, 250, 252-254, 256,
o(X,F)-
303
T(XF)- 168, 184 uniformly (norm) 97, 160, 161, 233, 256, 261, 263, 264, 9l, 142, 145, 148, 224, 225 97, 136, 160, 164, 169, 226,
weakly
258, 87 weekly*- (0--weakly) 160, 169, 226, 233, 257, 259, 261, 267, 285, 288, 294, 296, 299, 302, 312, 77, 80, 84, 113, 127, 274
47, 57, 58, 7l, 85, 305, 317, 319, 320, 339, 370, 375, 426, 431,
Weyl
240, 257, 261, 265, 267, 268, 299, 300, 4, 23, 57-61, 80, 86, 109, 111, 127, 138, 146, 150, 244-254, 282,
307
Commutant
canonical
expectation
150
Cone
Contraction
161, 169, 170, 177, 178, 194, 196, 9l Contour 321-330,440 Convolution 249, 252 Cook 8,277 Core 167, 182, 188, 191, 196, 267, 270, 274, 275, 277, 278, 282-284, 286, 287, 289, 290, 308, 35, 85, 96, 98105, 182, 191, 265, 277, 286, 340, 365
hard
360, 394, 443
Correlation
apex of
base of
causal
canonical
334
function(al)
104
hereditary 145 homogeneous 156 natural
39, 40, 218 93, 94
334
257,259 34, 36, 83, 334
convex
310
153, 220, 407, 460
306, 316, 405, 406
Cluster property
multiple
83, 84, 97, 104,
206, 290, 369, 370, 375 97, 160, 164, 169, 226, 232,
strongly
Christensen and Evans
dual
Connes and Takesaki
Holder 458-460
Choquet
(self-polar)
105, 145, 156 Connes 155, 156, 308, 310, 312, 252
Continuous
306
two-
self-dual
Construction, GNS
414
one-
105
pointed
Character
Chi
156
orientable
279, 280, 282, 284
9
102-108, 147, 155, 156, 226229, 232, 264, 299, 305, 369, 116, 157, 275
380-382, 391, 437, 443,
450-453
inequality
227, 222, 442, 443, 446, 454, 459
long-range 118 Courbage, Miracle and Robinson 218 Crossed product 136-141, 284, 310 skew
141
Cross section
290
Subject Cross-norm Cuntz
144
subset
46
46
45, 46, 86,
vector
109 et seq.
234
Dang Ngoc
and Guichardet
462
Decomposition 14, 32, 45, 105, 106, 117, 121, 216-218, 221, 275, 343, 379, 415, 424, in almost
periodic
290, 291, 300, 301, 144, 145, 244 everywhere defined 235, 267, 263, 306 implementation of a 269 inner 305, 144 nonclosable normal
430, 431, 433, 436,
437
barycentric 321, 324, 335, 350, 356, 367,447,458,461,^7, 114,301, 302
central
278-284, 293, 295, 383, 384 160, 233-249, 265-268, 282, 286, 292, 306, 307, 309, 4, 214, 244, 247, 249, 413, 427, 430 bounded 235, 236, 246, 249, 262, 263,
Derivation
85, 86
subspace
(factor)
317, 358, 370, 423,
440, 447, 460, 779, 143, 211, 227 sum 45, 356 direct integral 450, 453 ergodic 318, 374, 380, 386, 390, 393, 423, 429, 440, 461, 462, 463, 119,
norm-closed
theorem
strongly 203 uniformly 161, weakly* 203 Digernes 3 1 1 Dirac
Direct
sum
finite
Dixmier
319
infinity
317, 321, 373, 453, 454,
465
216, 219, 222, 236, 305, 369
orthogonal 36, 229, 319 polar 38, 72, 87, 89, 90, 92, 107, 108, 109, 112, 116, 130, 152, 232, 279, 287, 291, 292, 302, 312, 396, 397, 419, 27, 727, 275
positive linear functionals *-algebras 463
property of
a
lattice
over
336
theory
129, 317, 449, 450, 459, 460 317, 319, 373, 413, 414, 440,
459
Delbrück and Moliere Dell' Antonio and
435
9, 279
Doplicher Doplicher and
152, 312 ^29, 440, 446, 453 Doplicher and Kastler 463, 464 Dobrushin
Kastler and Robinson
and St0rmer
Douglas Dual
460
group
Duality
(see Group, dual) 126, 130, 169, 209, 439,
Dubin and Sewell
^59
Duhamel
function
two-point
442
93, 221,
225 Duneau and Souillard
Dunford
Dye 305 Dynamics, perturbed Dynamical flow
^59
155
97
159
13, 136, 137 136, 137, 139, 142, 310, 76-106, 204, 212-214, 235, 422, 423
System, C*275
461
462
48, 5l, 53, 62, 76, 130, 170, 203, 204, 216, 219, 305, 317, 350
System
218
Ruelle
439, 440, 450, 454 45, 46, 60, 439, J57, 363,
364
of
representations 317,440,446 spatial 439, 447, 449-451, 455, 465 spectral 95, 251, 320, 407
162
277
extremal
of States
159
Diflferentiable
integral
317, 318, 358, 359, 367, 440, 447, 727, 143, 144, 210
287
262
Direct
of
235-245, 306
spatial 268, 270-272, 274, Symmetrie 233, 144
144
Jordan
306
308
Diifeomorphism
direct
at
503
Density matrix 76, 85, 106, 124, 127131, 210, 405, 25, 36, 159, 268-274,
306, 226
Cyclic projector
Index
504
Subject
Index
Dynamical (Contd.) W^136-142, 310, 77-^7, 114, 123, 144, 150, 158, 165-170, 178, 212214, 237 Dynamics 159, 160, 4, 5, 45--62, 72, 78, 94, 119, 144, 176, 239, 250, 251, 259, 348, 349, 427 Dyson 252 Lieb and Simon and Lenard
Effros
441
447
465
Einstein
222
Extension
invertible
25, 38, 213
isometric normal
28 28
odd 121, 162 positive 7, 32, 38, 215, 276, 27, 92, 94,
116, 123
quasi-analytic 311 self-adjoint 28, 103, 147, 212, 213 unitary 28, 38, 72, 97, 107, 110, 111, 141, 214, 384, 395, 417, 418,411, 101, 102, 145, 146, 211-214 Elliott 152, 306, 309, 311 Ensemble, canonical 4, 437 canonical
microcanonical
4, 46, 437 4
89, 268-302, 432-435 conditional 268, 274-284, 295, 432, 434
maximal conditional
275
maximum
89, 90, 221, 267, 273, 274, 286-300, 340, 422, 435, 444 mean 290, 292, 296, 300-302, 348, 434 per unit volume
270
relative total
canonical
52, 58, 60
52
State
Extremal
point 53, 57, 317, 321, 325, 327, 330, 331, 350, 353, 405, 425, 464, 777, 720
Face
319, 321, 355, 369, 372, 373, 385, 428, 432, 436, 440, 447, 449, 450, 454, 455, 464, 722, 123, 131-136, 139, 230, 231, 339 stable 333, 334, 353, 354-356, 373, 377, 378, 301 Factor 7l, 127, 142, 151, 267, 277, 305, 310, 312, 355, 356, 358, 423, 446, 448
finite
134
hyperfinite infinite
137, 142, 150, 151, 305
148
Krieger 142, purely infinite semifinite
151
148
148
(see State, factor) 148, 355, 356 148 typel 148, 150 type II 148, 305 type IIi type 11^ 148, 150, 305 type III 134, 148-151, 421, 464, 132, State
Entropy
2(57-270,275-250 270,277
Envelope upper
225,226
Evans and Hanche-Olsen
analytic (entire) 97-101, 115, 178183, 231, 265-267, 272, 293, 76-80, 84, 94, 157, 158, 167, 207-210, 244-248, 277, 307, 334, 432 (see also Vector, analytic) even 121, 7(52 identity (see Identity)
lower
429, 442, 450, 457
Ergodic theory 13, 401, 423, 432, 435 (see also Action, Decomposition, State) Ergodicity, central 395 Evans
442
Element
grand
Equicontinuity 167, 185 Equicontinuous, T(X,F}- 168, 174, 204 Equivalence physical 14 quasi- 79-82, 128, 370, 395, 452 unitary 48, 56, 79, 80, 151, 283, 396,
type I
226
typellli
150, 151 150,151
type III l
150, 151,257
type IIIo
Ealk and Bruch
326, 327 326-328, 335, 338
227
Eannes and Verbeure 446
227, 222, 445,
Subject Fell
459
Feller
Fröhlich and Lieb
303
and Park
Fermi gas
4, 45-61, 76, 113, 119, 161, 175, 176, 197, 237, 354, 355, 387, 403, 415, 421, 423, 443 54
sea
surface
54, 176
Feynman 232, 452 Integrals 450 Feynman-Kac formula
66, 319, 321, 325, 327, 330, 336339, 349, 364, 37, 288, 290, 301,
345 almost
442
444
(magnetic) 304, 314, 315, 318, 330, 332, 336, 339, 345, 349, 438-442
relativistic
Fock
3, 461, 223, 409 197, 198, 201, 204
443
8, 797
Fock space Form
(see Space, Fock)
periodic 400, 401, 430, 435, 436, 464, 36 analytic 9l, 76, 79-83, 85, 97, 124 Borel 5, 295 concave 326, 328, 336, 99 convex 321, 325-327, 331-333, 353, 355, 377, 80, 94, 297, 299 countably additive set 321 lower semi-continuous 377, 37, 97, 108, 110, 159, 396 of positive type 463 subadditive 65, 329, 336, 337 superadditive 336, 337
bilinear
trigonometric
closable
upper semicontinuous
93, 94, 102, 18, 19, 23 26, 55 closed 26-30, 55 positive 26-30, 55, 73, 355 quadratic 26-30, 51-58, 73, 342, 356365, 444, 457
361,363 Symmetrie 77, 22 symplectic 23 sesquilinear 214, 270, 26, 40, 60, 78, 93, 334, 406, 436 Fortuin, Ginibre and Kasteleyn 442 Fourier analysis 25, 94, 241, 243, 244 transform(ation) 244, 250, 253, 257, 260, 417, 20, 49, 85, 97, 104, 161, 169, 170, 173, 179-181, 186, 205, 206, 253, 258, 335, 338, 349, 444 Free energy 270, 292, 295-295, 302, 347, 436, 440-443 mean
297
297, 295 277, 267
minimum Fresnel
integral
Friederichs extension
767
9, 277 104
430
326-328, 331, 332, 338, 27^, 275, 288, 301, 347,
365
Function(al) analysis Hermitian
152, 267 216-218, 222, 223, 305,
368
sum
equilibrium
305
affine
370, 372, 384,
external
Fisher
441
454
Simon and Spencer 440 Fujii, Furuta, and Matsumoto 313 Fukamuja 1 52 Misonou, and Takeda 460 Füll family of states 220, 221, 222,
385, 448, 449, 453
System
505
Function
Field(s) 10, 308 algebra 797, 198, 201, 218 construction theory 232, 440, Euclidean
Index
integration linear
555, 358, 394, 443, 454
42
multilinear
39
normal
76, 146, 384 number 36, 52, 396 positive (linear) 48, 5l, 76, 340, 346, 360, 368, 369, 425, 88 sesquilinear 57 tangent 174^176, 182, 234, 441, 443 truncated 39-44, 149, 151, 218, 219, 233
Wightman
463
Functionals
disjoint positive ordering of 52 Furuta
313
linear
370
145, 339, 370, 418,
297-299, 154, 182,
506
Subject
G-abelian
Index
381, 385, 386, 392-395, 397, 405, 414, 415, 418, 419, 427, 434, 462, 44, 45, 57, 133, 139-
403, 433, 144, Gr-abelian 433,
231, 339
414, 415, 417, 419, 420, 428, 434, 436, 455, 463 G-central 381, 387, 392-398, 403, 462, 720, 722, 2J7 Gallavotti and Pulvirenti and Mirade-Sole
311
438
Garding 9 and Wightman 22^ Garrison and Wong 227 Gauge automorphisms 797 invariance
197, 392, 402 transformations 39, 43, 202 Gaussian domination 440, 441 Gelfand 3, 7, 16, 152, 153
isomorphism 64 transform(ation) 63, 250, 126, 142 Generator (infinitesimal) 162, 165, 172-
305
Group 142, 401, 462, 463, 164, 169, 174, 202, 204-206, 233, 245, 248, 305 Q- 183, 202, 204-206, 233, 245, 247, 290
character drde
249, 250, 407
139
countable
142, 419
compact abelian cydic 413, 419 dual
142, 437, 455 137, 139, 165, 249, 407, 412, 428, 435, 455, 204
Eudidean finite
404
(abelian)
141, 465
two
generators
free
109, 110, 144, 146, 159, 164, 211, 222, 244, 245, 249
locally compact
168
strong
152
and Jaffe
Jaffe and Godement Golodets
...
on
gauge
304
Spencer
440, 422
463 252
Gorini, Kossakowski and Sudarshan
22^
Gram-Schmidt
orthogonalization procedure 442, 18 Graph (of an operator) 170, 186, 187, 188, 207
136, 463, 465, 752,
140
abelian 139, 249, 254, 310, 400, 401, 407, 408, 414, 417, 419-424, 429, 431-436, 455, 462464, 34
locally compact soluble 463 automorphism (see Modular automorphism) noncompact semisimple Lie 463 one-parameter 12, 13, 96, 97, 159, 160, 209-299, 4, 8, 77, 84, 86, 109,
modular
111, 114, 145-150, 209-212, 243247, 254, 282, 306, 310, 311, 427, 445^ 446^ 460
permutation 6 perturbed 247, 430, 404
quotient
424-430, 437
stabilizer
186, 187 278
187, 188, 281, 266, 371, 372
750
Poincare
second countable
Graph convergence Hubert space
463
797
locally compact
168
Generators, neighboring 202 Gibbs 12, 4, 430 condition 259-317, 342, 432, 433 State (see State, Gibbs) Ginibre 464, 390, 440, 442, 443, 453, 454 Glimm
430
discrete
206, 233-244, 265-279, 288, 290, 300, 304, 24, 37, 89-92, 96, 101,
weak
limit
Grothendieck
Co-
427
Mirade-Sole and Robinson
437, 438 86, 220, 338, 339, 354, 355, 373, 374, 380, 389, 395, 403, 404, 410, 411, 421, 449, 452, 459 compensating 374, 394 Griffiths 440, 442
Green's functions
amenable
436, 437
Miracle-Sole and Ruelle
Greenberg
455, 456, 465
413
symmetry 209, 404, 423, 424, 429 97, 189, 250, 253, 257, 261, 264,291, 301, 302,421,5,2^,50,
unitary
Index
Subject 50, 84, 88, 103, 130, 147, 150, 170, 180, 198, 334, 345, 409, 412, 413 of translations
Group
Hinterman and Merlini
444
120, 129
132, 173, 234, 422, 87, 96, 98, 162, 221, 334, 335, 416
231
152, 462
Guichardet
308
Inequality Bogoliubov 329,331,433 Cauchy-Schwarz 49, 50, 57, 78, 131,
442
Guerra, Rosen and Simon Guenin
Nakagami Indecomposability 14 Index set
251
459
Gruber
Ikunishi and
507
(see Correlation inequality) generalized Schwarz 213, 232, 222-224 Golden-Thompson 275
correlation
9, 1l, 13, 14, 156
Haag
Kastler and
Trych-Pohlmeyer
Hugenholtz
233
and Winnink
211, 220,
227
Trych-Pohlmeyer 233 Haagerup 155, 305, 311 Hamiltonian (see Operator, Hamiltonian) Hardy, Littlewood and Polya 460 Heisenberg 3, 223, 419 anisotropic model 243, 319, 333, 440, and
Peierls-Bogoliubov 275, 317 product 20 triangle 20, 132, 253, 272, 274 theory
model
Interaction,
56
norm
problem
220
441
364, 365, 447, 454
257
extemal 282
16
249
331, 332
242, 258, 260, 345, 427, 429,
Coulomb
257
3, 263
330, 331, 339, 346 241, 248, 251, 267, 437,
ferromagnetic finite ränge 438
437
Hille and
Phillips
Hoekman
233
Hohenberg Huang 220 Huber
surface
exponentially decreasing
304,306,311 441, 442
Hilbert's sixth Hill
across a
anti-ferromagnetic
energy
Hubert-Schmidt
319, 337, 440, 441
170
divergence
classical
243
243, 258, 334, 337, 338, 348, 350, 422, 424, 427, 440, 443
Higuchi
432
Infrared bounds
model
Herman
267, 435
Information
attractive
441
Isotropie
Helium
305
Kadison's
218
and Kastler
303
440
Heisenberg 335 Ising 257, 259, 321, 324, 333 long-range 261, 413, 414, 442 lower semi-bounded
nearest-neighbour
435
347, 438, 439 243, 248, 310, 313, 314,
13, 464, 227, 227
Hugenholtz
360, 366 319, 330, 332, 345,
one-body 319
Ideal
left
spin System 241, 244, 247, 250 positive 383, 388, 393, 401, 453 repulsive 220, 414, 461
right
stable
gas
45,46,56,421,426
Jordan
8
23, 40, 146 23, 39 two-sided 23, 24, 78, 133, 310, 352,
427, 454, 16, 84
Identity 7, 2l, 23, 25 et seq. approximate 23, 39, 40, 4l, 49, 56, 62, 78, 121, 153, 253, 353, 98, 99, 143, 184
of
a
364, 390, 394, 446
superstable 364, 453 translationally (or Z^-)
invariant
2^2,
257, 256, 290, 296, 300, 331, 338, 356, 364
243, 248, 251, 256, 338, 362, 364, 426, 422, 447
two-body
508
Subject
Index
Invariance condition
Involution
230, 232, 375
Irreducibility algebraic 1 54 topological 47 Ising
Kishimoto
156, 304, 306, 311, 231, 427, 439, 461
19, 42, 45
KMS condition
424, 438
model
243, 257, 319, 320, 329, 339, 340, 422-427, 439-443 Isometry (of a C*-algebra) 211 Isometry, partial 38, 110, 135, 292, 302, 371, 396
Isomorphism
13, 284, 5, 48, 59, 62, 76-102, 114-129, 145-153, 166, 167, 178, 210, 220-222, 259, 282, 294, 297-300, 307, 329, 405, 408, 422, 436 Kolmogorov-Sinai invariant 433 Köthe
154
Kramers and Wannier Kubo
43, 44, 78, 228, 341, 348, 355, 369, 385, 390, 405, 425, 433, 434, 444, 445, 452, 15, 19, 2l, 83,
Kurtz
111, 158, 175, 203
Landau and Lifschitz
anti-
211
Borel
Order Isotonic Israel
Kubo-Mari scalar
Sato and Ueno
410, 411
448
gas
448
Law of
243-347, 239-352, 422-427 large numbers 267
Lebowitz 435
Yang 439, 442 de Leeuw 303, 304, 458,
305
463
Lenz
304
Lieb 428
435
220
439
and Liebowitz
454
and Robinson
^27
and Ruskai
433
Schultz and Mattis Kac
220, 452 Kadison 78, 152, 154, 155, 305, 306, 309, 311 and
Ringrose 312 Baym 220 Kaplansky 152, 153, 305, Kastler 14, 461^64, 233,
Katz
Kelley Kernel
234
459
MacGibbon
303
458
Mackey 16, 465 Magnetism 424 ferro-
152
para-
442
318, 334, 353 306, 315, 318, 333, 337, 424 306, 314
Magnetization, spontaneous
43, 79, 151, 83, 117, 373, 371, 373, 446
Salzburg
Phillips
anti-ferro-
232
Kirkwood and
Linear response 95 Lindenstrauss, Olsen and Sternfeld
307
152, 304, 233, 446
and Sherman
^22
22^
Lumer and
461, 465
Robinson and Ruelle Kato
Lindblad
464
Kadanoff and
and Robinson
459
424
Lewis and Pule
7
and Wigner 277, J0rgensen 304, 459 Jost
442
Lee and
Jacobson and Rickart
Jordan
436
462, 431, 432, 441 Laplace transform 166, 169, 276 Lattice 323-325, 108, 239-350, 419427
Itö and McKean
Jaffe
220
and Ruelle
family (of *-algebras)
Jadczyk
93
152, 227
and Robinson
434
Jacobs
product
304
Lanford
295, 435, 465 211,213 211,213,229
Jordan
436-439
13, 220
437, 438
306, 318, 320, 331, 334, 336, 337, 345 Manifold, differentiable 159 Manuceau
225
Subject
509
13, 83, 318, 319, 335, 359, 387, 424, .434, 458, 96, 135, 266,
probability
Map 210, 216, 218, 221, 222, 369, 387, 464, 162 Borel 294, 295, 298, 299, 456
affine
341
364-366
faithful
219
invertible
209, 211, 215, 219, 229, 232, 305, 341, 222, 225 quotient 295, 424, 456 restriction 357, 358, 460
positive
Martin
Schwinger 220 255, 255, 308, 311, 312,
Matrix units
Maximum modulus
principle
79, 8l
and Montroll
435
379, 388, 400, 462, 163 invariant 280, 302, 313, 400, 414, 415, 421, 422, 429, 435, 436, 455, 457, 463, 36, 44, 52, 163, 193, 256, 296
Measure
213, 322, 375, 433, 440, 450, 88, 318
barycentric
97, 103, 321, 322, 340, 341, 350, 357, 370, 452, 187, 368, 369, 375,
444
of bounded Variation
97
370-372, 460, 461
conditional... Wiener
(or point)
368
319, 3.22, 323, 343,
338, 456 ergodic 14, 433 575, 577, 380
functional
Gibbs
414, 419, 432, 433, 458, 779 367-379, 386, 387, 448-450 Mechanical work 272, 275, 216 159, 287, 288, 289, 311, 555
classical matrix
3
3, 5, 6, 16, 121, 154, 159, 5, 277, 257 statistical 6, 10, 13, 16, 84, 124, 129, 273, 306, 311,461,5,^, 775,277,
quantum
3
wave
Borel
Gaussian
370-372, 390, 722
unique maximal 318, 334, 335, 337, 338, 363, 366, 377, 382, 385, 390,
257
91,96
Dirac
365
440^49, 452, 455, 465
Mechanics
307
Mean
central
75, 184
signed 323 simplicial 343, 364,
Wiener
435
Mcintosh
5, 250, 261, 407,
350
ö--finite
subcentral 343
313
Matsumoto
Baire
Riesz
Standard
13
Mayer
369
projection-valued
408, 421, 188 Radon 321, 322, 326, 347, 356, 425, 88, 124, 367-369, 375 regulär 321, 322, 332, 340, 341
223-229
completely positive contractive
and
Index
456
445
Wagner 440 Messager and Miracle-Sole 441 Mini-max principle (theorem) 55, 365, 366, 393, 394, 448 Mermin and
Minkowski space 460
369
Misonou
14, 393, 4l, 177, 422 401-403, 42, 62, 320 of all Orders 402, 403 strong weak 423, 455, 465
Mixing
strong
...
Miyadera
303
Modular
86, 96, 97, 102, 147, 149, 230, 278, 279, 281, 285, 84, 114, 124-128, 151, 227,
137, 249, 289, 428, 435, 437, 438, 455, 463, 757, 142, 211, 217 Lebesgue 380, 5l, 182, 191 maximal 318, 325-327, 330-338, 343, 355, 356, 359, 361, 363, 377, 386,
automorphism (group)
424, 777, 120-122, 131, 136 maximal orthogonal 339, 359, 361, 378, 395, 440, 460 orthogonal 320, 339-462, 777, 143
conjugation (involution)
Haar
277, 430 condition
83
89, 94, 102, 106, 226, 227, 278, 391, 397, 707, 775, 725, 160, 194, 283
function
137
510
Subject Index
Modular
(Contd.)
Observables, algebra
operator 89, 9l, 94, 96, 102, 103, 226, 227, 229, 278, 283, 285, 391, 403,
86, 107, 118, 125, 160, 161, 187, 194, 283, 404 relative
operator
...
theory Modulus
(of
an
Mokobodski M011er
275, 257
83, 78, 85
element)
34, 38
458
morphism
162, 167, 169, 175,
192, 195 matrix
163, 169, 170, 173, 233, 236 4, 36, 53, 54, 62, 346, 350 438
Morphism
42, 213, 224, 227, 228, 341, 342, 390, 450, 451, 162-166
anti-
211,213,224,227,228
Jordan
209,211 3, 1l, 16, 154
Miirray
Nagel 463 Nagumo 303 Nagy 152, 304 Naimark
3, 7, 16, 152, 153, 305
Nakagami Nambu
308
Verbeure and Weder
Nearest-neighbour(s)
227
227, 225
327-331, 421,
426, 448 Nelson
304, 448 Net 39, 119, 123, 323, 326, 336, 337, 342, 379, 383, 386, 388, 392, 396, 403, 415, 463, 108, 122, 240 decreasing 118, 119 increasing 76, 78, 79, 130, 146, 155, 348, 30, 72, 281 Net of subalgebras 118, 121, 123 Neumann series 26, 171, 241, 268, 274 233
Novikov
72
449, 451
Norm
20
Normal
subgroup
Olsen
226
464
Onsager 439, -Yang 439 Operator adjoint 19
448
affiliated with
an
algebra
87, 90, 287,
annihilation
5-72, 77, 25-27, 37, 45, 58, 148, 161, 217, 226, 258, 398, 415, 425, 426, 444 antilinear 86, 88, 16 antiunitary 84, 87, 89, 210, 226, 248 bounded 20, 160, 173, 197, 203, 205, 301, 348, 378, 409, 415, 418, 7, 77, 14, 16, 33, 87, 145 closable 88, 177, 187, 235, 267, 271, 272, 276, 5, 13, 24 closed 87, 9l, 170, 171, 177, 197, 235, 276, 348, 25 compact 2l, 72, 129-131, 210 84
174, 265 252, 257, 5-72, 77, 25-27, 45, 58, 148, 161, 217, 226, 258, 398, 425, 426, 445
creation
decomposable 443-445, 448, 453 diagonalizable 443-448, 451, 455 dissipative 174-177, 193, 196, 197, 234, 235, 290, 303, 304, 306, 96, 2^5
double difiference
349
essentially self-adjoint
272, 278, 289,
5, 72 finite-rank
210
Hamiltonian
4, 12, 159, 163, 273, 4, 36, 45-48, 52, 57, 65, 151, 196, 241, 349-365, 390, 411, 426, 428, 450, 455
Laplacian 424, 428, 429, 440,
463
188, 48, 49, 54, 60, 78, 355, 370, 372, 426, 445, 446
lower semi-bounded
5, 57, 6l, 63, 169, 351, 361-363, 410
Observable
5, 6, 7, 12, 160, 209, 370, 461, 462, 3, 6
algebra
309
and Pedersen
conservative
Naudts and Verbeure
Nondegenerate
Olesen
conjugation
427
Newton
12, 122, 141,
116
Momentum Montroll
of
159, 4, 197, 198
137
lowering modular
252
(see Modular operator)
negative part
of
an
34
Subject
4, 7, 10, 25-37, 218, 363, 425,
number
429, 442 57, 76, 89, 104, 273, 351, 2629, 3l, 40, 60, 73, 78, 114, 135, 208,
Path
367-376, 390, 452
Pauli
(exclusion) prineiple
184, 240, 269-271, 273, 277, 280, 286, 291, 25-30, 36, 40-48, 57-60, 73, 76, 78, 84, 104, 276, 341, 352, 356-366, 410, 431 shift 252,257
skew-adjoint 292 spin raising 257, 255, 579, 25, 428 spin lowering 257, 255, 319, 425, 428 Symmetrie 104, 182, 238, 240, 246, 269, 272, 278, 8, 13, 54, 208, 357, 423, 428 traee-elass 6, 68, 76, 129, 5, 46-48, 57-59, 219, 220, 363-365, 410, 411 unbounded 5, 153, 463, 60, 116 unitary 8, 47, 57, 159, 210, 225-227, 240, 247, 248, 269, 293, 295, 374, 442, 13, 19, 2l, 6l, 90, 127, 142, 164, 406, 407 Weyl 8, 1l, 14, 18, 19, 30, 35, 57-60, 73, 417, 442 (see also Element) Operators, measurable family of 442, 443
Orbit
141, 142, 425, 429, 430, 434-438,
Peierls
152, 305, 309 415, 439, 440
argument 319, 439, 440, 445 Period, approximate (or 8-) 430 Perturbation 160, 171, 184, 193, 202, 246, 290, 95, 145-149, 164-169, 176, 194, 196, 212-215, 261-263, 273, 316, 333, 350, 358-363, 446
193, 207, 363 195, 196, 311, 94, 146, 149, 150, 153157, 165, 232, 308-314 Phase 10, 57, 339 374, 462, 113, 118, 119, 144, 289, pure
relatively
bounded
Perturbation series
(expansion)
301, 304, 422, 445
3, 345, 426, 443
Separation 288 space transition
10, 3, 237, 306, 324, 338, 352, 404, 439-445
Phillips
303
Planck's constant Polarization
4
identity
38, 67, 182
Potential Coulomb
559, 448 positive 364, 394, 395, 459 Potential energy 356-359, 386, 424 finite ränge
394
358, 359, 458
559, 360 translationally invariant 555, 590, two-body 555, 590, 595, 414 Lennard-Jones
Order
anti-symmetrie
39, 325
325
reflexive
39, 36, 39, 318, 325-327, 334336, 346, 348, 349, 359, 424, 458, 459, 27
relation
structure
transitive Ornstein
464
Powers
156, 306, 225
and Sakai
39, 325 relation
Projeetion final finite initial
7
equality
Partial order
325
311
450-454
12
36
306
Parseval's
394
69, 75, 83, 216, 233, 280, 294, 299, 446 Pressure 55, 54, 270, 387, 393, 443, 445,
435
Oscillator, harmonic
Poulsen
Predual
78
Orthogonality
Paige
194, 202, 204
bounded
/c-body
165
Ota
337, 241, 310, 319
matrices Pedersen
positive part of an 34 raising 252 self-adjoint 5, 6, 89, 104, 153, 182,
9, 12, 447,
454
positive
341
511
277
Passive System
292
normal
Index
245
infinite
110
147 110 147
Projections, equivalent
147
512
Subject
Index
Propagation velocity 252-257 Pseudosupported 318, 322, 331, 367, 372, 373, 377, 138 Pulvirenti
direct
311
Pusz and Woronowicz
232, 237, 429
Radius of
analyticity 178 Radon-Nikodym derivative 90 Range, numerical 292, 302, 312 Range (of an interaction) 242, 258 finite
Ray
258
density matrices,
381-400, 414,
443, 450 truncated
393
Reed and Simon
152, 304, 452
Reflexive Danach space
Regularization Representation
of
449, 450 8l, 319, 339
of
sum
disjoint 370 equivalent 48, 370, 135, 139, inequivalent 9, 168, 170, 218 measurable family of 447 Representative 43 Resolvent
25, 27, 9l, 165, 168-170, 184, 274, 306, 311, 775
206
Rickart
Lebesgue lemma
Rideau
219
Rieffei
155
Riemann
approximant (sum)
integral 57, 55, 55, 132,
Nagy
Robbins
431
covariant
Roberts
304, 311, 312
cyclic
Robertson
349
136-138, 133, 204 14, 45, 56, 57, 60, 353, 358, 375, 60, 82, 84, 88, 130, 139, 170, 179, 335, 405
factor
(see Factor) 8, 14, 43, 44, 79, 133, 138, 139, 220, 237, 238, 272, 310, 352, 110, 199, 204, 210 Fock-Cook 8, 1l, 25, 30, 33-37, 52, 170, 217-220, 350, 446 155, 374 group irreducible 8, 47, 57, 80, 151, 211, 224, 310, 352, 368, 393, 394, 448, 463, 34, 142, 168, 204, 218 faithful
8
multiplier 20 non-degenerate normal
45, 80, 446-448
79
sub-
43
44, 8l
uniformly bounded 254, 256 unitary (group) 3, 5, 250, 251, 379, 383, 416, 463, 169,
152, 304
305
Robinson
304-307, 311, 461, 464, 219, 220, 232, 427, 434, 440
and Ruelle Rocca and
Root
434
Sirugue
Roepstorff
227
227 square deviation
mean
349
Rudin
152, 227 Ruelle 9, 459-465, 22P, 436-443, 446, 453, 459 Ruskai
459
Russo and
Sakai
regulär 2l, 33-36, 148, 204, 208 Schrödinger 6, 8, 9, 34-36, 435 space
167, 243,
33, 215, 243
Riesz and
a
57
56, 63
42, 23 state
305
Riemann
308
associated with
isometric
140
184, 186, 188 convergence Return to equilibrium 759, 777, 196
159, 210
Reduced
Representations direct integral
Dye
305
76, 152, 155, 306, 307, 309, 311, 312, 459, 460
Saturation problem 304 Scattering theory 163, 164, 233, Schrödinger 3, 6 equation 4, 46, 376 Schultz
454
8
Schur's lemma
Schwartz
153
152
Schwarz reflection
principle
400, 403, 407-409, 412, 415, 419, 421, 422, 426, 429, 430, 464, 6, 7, 20, 98, 129, 141, 144,
Schwinger
13
Sector of
operator
204
Second
an
Second law of
275, 8l
292
thermodynamics quantization 8, 45, 233
272
Subject Sector of
an
operator
Simon
287
Segal 7-9, 16, 152, 153, 461, 225, 22^ Semigroup 161, 164-168, 203, 9l, 144, 223, 284, 361, 368, 370, 373, 378, 383, 446 Co- 164, 169-178, 187-194, 202, 235, 303, 304 q- 164, 170, 171, 183, 193-196, 202, 205, 303, 304, 225 completely positive 22^, 225 dual
of
65, 66, 163, 168, 174
138
C*-
339, 441
i(X.F)- 204 Separability condition
S 352, 353, 367, 372, 377, 378, 426-428, 432, 436, 440, 447, 449, 450, 454, 455,
Poulsen
Separating 85
80, 86, 88, 109, 226-230, 277289, 311-444, 82, 84, 86, 87, 104, 107, 114, 120, 153, 183, 190, 263, 275, 276, 279, 409, 411
vector
andWinnink
analytic 295, 357, 458, 460 Baire (see Baire set) Borel (see Borel set) convex 53, 59, 66, 317-359, 293 directed 39, 46, 121, 123, 324, 331,
458, 460
Skau
218
449 3
Space analytic Borel 295 configuration 120 conjugate 70, 238 direct integral 439-442, direct
322, 357, 372, 373 322, 331-333, 338, 357, 458,
Fock
7, 9, 12, 15, 30, 34, 46, 56, 349, 415, 444
quotient 23 polish 295,440,456 measurable family of 441, 442, 444, 447, 449
Spectral 253
concentration
metrizable
,
204
7, 249, 281, 298, 308, 50, 99, 102, 188, 204 252
values
25-28, 32, 35, 5l, 6l, 62, 142, 149, 165, 220, 240, 245, 248, 251, 252, 256, 257, 263, 294, 306, 309, 407, 419, 420, 434, 464, 27, 33, 84, 104, 141, 147, 171, 172, 176, 180, 181, 183, 190, 351, 352, 365,
132, 133, 318, 331, 333,
339, 458
/z-measurable 356, 357, 452, /^-negligible 356, 357, 440 pseudosupporting 322, 331
454
366
resolvent
25, 28 331, 342 supporting 322, 357, 426 Sewell 227, 222, 436, 437 stable
Stinespring
219
Shannon and Weaver
435
Sherman
58, 7l, 87, 104,
132, 135, 118 radius 25, 26, 29, 3l, 37, 44, 59, 241, 312, 22 subspace 25 1 252, 257, 259, 410, 142,
Spectrum
133
Shale and
449
439
sum
theory
345
GS-
226
227,^55
projector (projection)
Set
F^-
464
and Testard
Sirugue
460
subset
464
Bauer
Sommerfeld
184-192
Seminorm
152, 304 334, 335, 367, 368, 385-387, 405, 406, 458, 464, 117, 118, 122, 131, 133, 134, 139, 230, 231, 301, 306,
Smoluchowski
positively preserving 222 Semigroups, convergence
513
Simplex
Slawny
303
Index
460
of
an
abelian
Arveson r
C*-algebra
6l
310
310
407, 412-415, 419-421, 424, 436, 455, 457, 464, 163
point
semi-bounded
261
514
Subject
Spin
Index
7, 239, 241, 424
311, 120, 211, 220, 222, 236353, 413, 422
168-176, 192, 194, 197, 226, 227, 338-351, 443
System
(G-) invariant
13, 117, 240, 268, 272, 311, 374-387, 390, 393, 398, 400, 402-407, 414, 415, 423, 427, 431, 432, 434, 459, 461, 462, 44, 52, 77-88, 98-216, 287, 290, 294, 301306, 315, 333, 336, 345-350, 422,
350-352
waves
Square root 32, 34, 36, 152, 235, 236, Stability 13, 160, 184, 202, 89, 90,
306
99, 145, 148, 150, 158, 164-183, 192-196, 211, 232, 233, 340, 364, 390, 410, 423, 447, 454 local
thermodynamic
275, 437 Standard form 83, 155, 226, 228, 305 State 42, 48, 5l, 153, 344, 345, 346, 350, 352, 353, 358, 361, 363, 439, 452, 23 almost periodic 414, 424, 430-433, 422 analytic 39, 40, 382, 397, 402 bound
chaotic
locally
normal 118, 124, 125, 129, 133, 219, 317, 352, 356, 373, 460, 25, 34, 52, 62, 170, 395, 396, 401, 415, 420
197
97
443
(G-} ergodic 374, 377, 401, 405, 407, 408, 420, 423, 424, 428, 464, 131-135, 138,
378, 393-398, 410, 413, 418, 440, 452, 459141-144, 240,
303
6
7, 9, 75-83, 96, 129, 131, 210, 219, 220, 238, 299, 355, 370, 371, 383, 390, 398, 452, 460, 462, 25, 30-36, 76, 80, 115, 123-126, 132, 133, 151, 169, 170, 174-176, 276, 337-340, 341-344 passive 90, 101, 102, 211, 212, 215 periodic 414, 141, 144, 210, 211, 290, 293, 417, 422 perturbed 757, 759, 174-177, 194, 211, 265, 276, 279, 316 physical 7, 122
primary
318, 727 factor 8l, 128, 317, 356, 358, 372, 401, 402, 403, 452, 460, 117-120, 133, 134, 139, 140, 151, 160, 184, 199, 219, 321 faithful 13, 83, 85, 96, 299, 403, 84, 115, 116, 124-128, 264, 275-279 36, 170
24, 219
guage invariant
44, 48-50, 54, 59, 60, 68, 78, 100, 108, 169, 206, 210, 219,
348
Gibbs
81
6, 53, 57, 58, 62, 352, 356, 358, 359, 405, 452, 461, 133, 167, 172, 174, 175,
153, 343, 350367, 368, 372, 134, 141-146, 181, 273, 275 quasi-free 40, 43, 44, 48-52, 54, 59, 60, 69, 78, 100,108, 169, 170, 219, 233, 388 regulär 23,24,29,33-37 trace 84, 87, 311, 421, 76, 83, 97, 102, 109-112, 121, 123, 154, 174, 184, 194, 211, 307 vector 49, 131, 356, 395, 133, 153, 160 pure
extremal G-invariant
density
mixed normal
completely passive 101, 226 even 43, 103 equilibrium 1l, 12, 273, 3, 5, 40, 4775, 88, 112, 118, 123, 148, 149, 158, 159, 176, 177, 198, 211, 218, 219, 237-239, 260-263, 286, 291-294, 300-315, 354-363, 400, 422, 436-
Fock
285, 5, ^5-57, 75-84, 88, 92, 95, 97-216, 221, 225, 237, 260-264, 295-297, 306, 308-320, 330-340, 353, 354, 423, 437-441
167
ceiling 97, 108, 113, 158, 168, 192, centrally ergodic 395, 396, 202
finite
436
KMS-
States
4, 46-50, 57-77, 88-93, 121, 144, 260-275, 298, 315-339, 354
disjoint
et seq.
neighbouring 219 orthogonal 339, 340
ground 9, 273, 277, 289, 309, 36, 54, 97-113, 131-141, 145, 158, 164,
9, 370, 395, 777, 725, 7597^7, 196
set of
7, 53, 59, 6l
Subject
Boltzmann
384, 387, 391, 392, 395,
447, 450 121, 7, 555, 366, 383-394 Fermi 121, 156, 7, 355, 365, 366, 384, Böse
57, 220 53,340,347 Tensor product 142-145, 7, 259, 416
Sternfeld
464
Stinespring 305 Stirling bounds number
zero
Theorem
Alaoglu 98 Alaoglu-Bourbaki
53, 68, 154, 163, 168, 176, 205 bicommutant 72, 445
390, 394
Bochner
431
8, 16, 305, 462, 464, 279
Streater
427
capacity 460 Caratheodory-Minkowski Carlson's
and
Wightman 231 Subadditivity 277, 272-275,
central
19
fixed point Subspace cyclic 46
cocycle 257, 252 Radon-Nikodym Connes-Takesaki duality Connes'
measurable
stable
derivation 441
edge
377, 427, 432, 435, 436, 437, 453 finite
322-330, 335, 337, 361, 363, 371, 377, 432, 97, 775, 722
Susceptibility, magnetic 318 Symmetry 136, 209, 374, 423, 424, 7^7, 239, 286 inner
374, 413, 423, 779 795
spin reversal 527, 324, 325 Wigner 210, 225, 226, 248, 305, System, closed 267, 294 isolated open
of the
Effros'
44
309 Summable central sequence Support of a measure 318, 322, 376,
broken
307
267
262, 224 8l, 83, 84, 99, 153
wedge
450
Verbeure
725
Fröhlich-Pfister
557, 441 257, 258, 757
Fubini's
Goldstone's
176
140, 231 Hahn-Banach 59-61, 65, 66, 75, 101, 154, 174, 200, 218, 221, 325, 328, 329, 339, 360, 403 Hille-Yosida 171, 173, 174, 178, 181 Kadison's (transitivity) 154, 402, 777
Haag's
Kaplansky density
74, 103, 122, 123, 154, 218, 232, 236, 347 446
Kovacs-Szücs
460
13, 83, 155, 310, 83, 221, 227,
ergodic 383, 724 53, 59, 6l, 153, 217,
mean
Krein-Milman Takeda
147 149
Fannes-Vanheuverzwijn-
Kato's
263,291,294
Takesaki
235
Connes'
137, 149
44
invariant
460, 722
128
Connes'
124
321
9l, 155
Cartier-Fell-Meyer Cauchy 88 closed graph 207,
288
271,288,289,433
Subalgebra
261, 263, 269, 309,
421
St0rmer
strong
103
Borchers-Arveson
401
5, 16
Support,
515
negative
Statistics
Stone
Index
163, 173, 201 98, 155, 341
Krein-Smulian
216, 220, 221, 262, 268, 306, 307,
dominated covergence 100, 185, 186, 412, 428, 437, 86, 87, 148, 156, 166, 173, 179, 181, 184, 185, 215, 371, 372, 401 Liouville's 275, 79, 99, 775
315, 318, 422, 428, 439-441 306, 318, 439, 441
Lumer-Phillips Mackey's 98
228
and
Lebesgue 459
Tomiyama Taylor series 34 Temperature 3, 5, 10, 46, 57, 70, 128,
critical
177
516
Subject
Index
Theorem
(Contd.) Mackey-Arens
Tomita-Takesaki 223
Markov-Kakutani fixed
point 415 403, 464 mean ergodic 378, 383, 386, 391, 394, 398^00, 411, 415, 421, 422, 462, 44, 162 Mazur's
Minlos'
455
monotone convergence
Parseval's
theory
13, 83, 84, 9l,
146, 84, 104
327
Tomiyama
459
property E of
150
Topology discrete
435
locally convex 59, 65, 97, 163 locally uniform 132, 133 Mackey (i:(X,F}-} 98, 155, 163, 164,
180
167, 169, 174, 184
Perron-Frobenius
431
metric
Phragmen-Lindelöf 80, 99 Pontryagin's duality 139
20, 132, 133
quotient
295
213, 792 Riesz representation 68, 70, 83, 213, 239, 322, 350, 124, 368, 375 Roepstorff-Araki-Sewell 9l Roepstorff-Fannes-Verbeure 95 Sakai's 76, 233 spectral 5 spectral mapping 3 1 SNAG (Stone-Naimark-Ambrose-
66, 70, 74, 108, 295, 7^5, 275 a-strong 66, 70, 74 strong* 69, 70, 74, 108, 295 (j-strong* 69, 70, 74 a(X, F}- 97, 98, 99, 163, 166, 171 ^-weak 67, 70, 74, 78, 279, 445, 146 uniform (norm) 20, 25, 53, 70, 108, 127, 131, 163, 169, 194,289, 335, 707, 272, 306, 436 weak 67, 70, 74, 108, 295, 356, 790 strong operator 127, 309, 750, 205,
Godement) 250 Stinespring's 223
weak*(ö-(U*,U)-)
Pusz-Woronowicz
strong
101
Radon-Nikodym
Stone's
275
251
Stone-von Neumann
uniqueness
6,
33, 34, 218 Stone-Weierstrass
24, 63, 64, 325, 346, 347, 376, 387, 202, 207,
368 Tauberian
Taylor's three-line Tomita's
250, 253 398-404
Tomita-Takesaki
94, 155, 311, 123
304
uniform boundedness
100, 176 von Neumann density 72-74, 154 Thermal wavelength 55, 55, 65 Thermodynamic limit 10, 1l, 13, 5, ^9-57, 63, 67, 74, 75, 86, 109, 220, 237, 260, 298, 366, 372, 381-383, 391-395, 401, 417, 423, 438, 442, 454
Thomas Time
development
Tomita
faithful
148, 149 148, 149 semifinite 148, 149
normal
Trajectory 367-381, 384, 386, 387, elementary 385, 386, 390 composite 385, 391 Transfer matrix
437-440
space
time Trotter
13, 41-44, 57, 62 57, 62 304
304, 757, 225, 370
160
227
13, 83, 459, 460, 227
394
Translations
product formula Twist 206, 302
443
Titchmarsh
181, 355, 116, 302,
68
norm
80-82, 152, 155, 157, 274 341, 450, 452
Trotter-Kato
14, 53, 68, 131, 132, 206, 307, 329, 335, 337, 377, 406, 86, 108, 109, 128, 169, 272-275, 288, 340, 348, 411, 436 weak operator 3, 341, 230 Trace 145, 280, 281, 287, 46, 191, 246, 265, 269, 281-283, 301
176, 346, 114, 298,
Uncertainty 432, Urysohn's lemma
433 332
Subject Vacuum
VanDaele van
9, 37, 4l, 44 155,220
der Waerden
16
152
Vaught Vector
almost
periodic 408, 414, 431 analytic 5, 24, 38 cyclic (see cyclic vector) entire analytic 38 separating (see Separating vector) von
Neumann
Wannier
517
6, 218, 448
Weyl
criterion
253
Wick
product 444, 456 Wiener integral 366, 367, 383, 453 Wightman 9, 16, 463 Wigner 7 (see also Symmetry, Wigner) Wils
460,579
Winnink
13, 227
Woronowicz
155
X-Y model
243, 257-261, 319, 337, 339, 426, 427, 441
440
145, 146
Weight dual
3, 5, 1l, 16, 154, 465, 218
Index
311
faithful normal
semifinite
Yaglom Yosida
146, 147, 149 146, 147, 149 146-149
and
Yaglom
433
152, 303, 464
Yngvason
463
Zorn's lemma
46, 77, 326, 332, 441, 458
